The New Oxford Annotated Bible has this to say in its annotations for verses 33b-36:
Many scholars regard this passage as a later non-Pauline addition, because it disrupts the flow of the argument from v. 33a to v. 37; it contradicts the assumption of 11.5 that women will prat and prophesy in the assembly; it resembles the viewpoint of the Deutero-Pauline letters (see 1 Tim 2.9-15); it exhibits non-Pauline sentiments, e.g. v. 34b, as the law also says, and vv. 34-35 appear after 14.40 in some manuscripts.
Bart Ehrman also mentions these verses in chapter 7 of his book, Misquoting Jesus: the Story Behind who Changed the Bible and Why.
The authority on the authenticity of the verse is Philip Payne, who argues from ms. evidence that the verse is an interpolation. Philip B. Payne, “Fuldensis, Sigla for Variants in Vaticanus, and 1 Cor 14.34-5,” NTS 41 (1995) 240-50, 261.
NETBible footnote is as follows:
Some scholars have argued that vv. 34-35 should be excised from the text (principally G. D. Fee, First Corinthians [NICNT], 697-710; P. B. Payne, “Fuldensis, Sigla for Variants in Vaticanus, and 1 Cor 14.34-5, ” NTS 41 [1995]: 240-262). This is because the Western witnesses (D F G ar b vgmsAmbst) have these verses after v. 40, while the rest of the tradition retains them here. There are no mss that omit the verses. Why, then, would some scholars wish to excise the verses? Because they believe that this best explains how they could end up in two different locations, that is to say, that the verses got into the text by way of a very early gloss added in the margin. Most scribes put the gloss after v. 33; others, not knowing where they should go, put them at the end of the chapter. Fee points out that “Those who wish to maintain the authenticity of these verses must at least offer an adequateanswer as to how this arrangement came into existence if Paul wrote them originally as our vv. 34-35” (First Corinthians [NICNT], 700). In a footnote he adds, “The point is that if it were already in the text after v. 33, there is no reason for a copyist to make such a radical transposition.” Although it is not our intention to interact with proponents of the shorter text in any detail here, a couple of points ought to be made.
(1) Since these verses occur in all witnesses to 1 Corinthians, to argue that they are not original means that they must have crept into the text at the earliest stage of transmission. How early? Earlier than when the pericope adulterae (John 7:53-8:11) made its way into the text (late 2nd, early 3rd century?), earlier than the longer ending of Mark (16:9-20) was produced (early 2nd century?), and earlier than even “in Ephesus” was added to Eph 1:1 (upon reception of the letter by the first church to which it came, the church at Ephesus)—because in these other, similar places, the earliest witnesses do not add the words. This text thus stands as remarkable, unique. Indeed, since all the witnesses have the words, the evidence points to them as having been inserted into the originaldocument. Who would have done such a thing? And, further, why would scribes have regarded it as original since it was obviously added in the margin? This leads to our second point.
(2) Following a suggestion made by E. E. Ellis (“The Silenced Wives of Corinth (I Cor. 14:34-5),” New Testament Textual Criticism: Its Significance for Exegesis, 213-20 [the suggestion comes at the end of the article, almost as an afterthought]), it is likely that Paul himself added the words in the margin. Since it was so much material to add, Paul could have squelched any suspicions by indicating that the words were his (e.g., by adding his name or some other means [cf. 2 Thess 3:17]). This way no scribe would think that the material was inauthentic. (Incidentally, this is unlike the textual problem at Rom 5:1, for there only one letter was at stake; hence, scribes would easily have thought that the “text” reading was original. And Paul would hardly be expected to add his signature for one letter.)
(3) What then is to account for the uniform Western tradition of having the verses at the end of the chapter? Our conjecture (and that is all it is) is that the scribe of the Western Vorlage could no longer read where the verses were to be added (any marginal arrows or other directional device could have been smudged), but, recognizing that this was part of the autographic text, felt compelled to put it somewhere. The least offensive place would have been at the end of the material on church conduct (end of chapter 14), before the instructions about the resurrection began. Although there were no chapter divisions in the earliest period of copying, scribes could still detect thought breaks (note the usage in the earliest papyri).
(4) The very location of the verses in the Western tradition argues strongly that Paul both authored vv. 34-35 and that they were originally part of the margin of the text. Otherwise, one has a difficulty explaining why no scribe seemed to have hinted that these verses might be inauthentic (the scribal sigla of codex B, as noticed by Payne, can be interpreted otherwise than as an indication of inauthenticity [cf. J. E. Miller, “Some Observations on the Text-Critical Function of the Umlauts in Vaticanus, with Special Attention to 1 Corinthians 14.34-35, ” JSNT 26 [2003]: 217-36.). There are apparently no mss that have an asterisk or obelisk in the margin. Yet in other places in the NT where scribes doubted the authenticity of the clauses before them, they often noted their protest with an asterisk or obelisk. We are thus compelled to regard the words as original, and as belonging where they are in the text above.
Interpolation: Strengths and Weaknesses
To recap the main point of the last section: if 1 Corinthians 14.34–35 clearly envisions a blanket prohibition of women’s speech in ecclesiastical contexts; and in context this certainly related to and included their prophesying. Yet this appears to blatantly contradict 1 Corinthians 11, where women are assumed, without any critical comment, to engage in prophesying in ecclesiastical contexts. (Cf. 11.16; Fitzmyer, 412.) And although the validity of the notion of Pauline theological or explanatory inconsistency — even in the same letter — has been amply demonstrated elsewhere,[9] the other oddities of the injunction is 1 Corinthians 14.34–35 in relation to this other Pauline thought still calls for a more immediate explanation.

As mentioned earlier, the dominant scholarly position today is simply to excise these verses altogether.

And on one hand, this gives us a quite satisfactory solution to the problem. Relying on insights from Graham Clarke’s article “‘As in All the Churches of the Saints’ (1 Corinthians 14.33),” which suggests a persuasive way of parsing the verse leading up to v. 34, we can read 14.30-33, 36 together as
If a revelation is made to someone else sitting nearby, let the first person be silent. For you can all prophesy one by one, so that all may learn and all be encouraged; and the spirits of prophets are subject to the prophets (for God is a God not of disorder but of peace), as in all the churches of the saints. Or did the word of God originate with you [ὑμῶν]? Or are you [ὑμᾶς] the only ones it has reached? (Modified NRSV)
encouraged to “yield the floor” to another who receives a revelation; and in the final line Paul delivers a sort of biting sarcastic remark aimed at those who might not properly do this. (For the sake of brevity, I’ll skip over some further thoughts I had about the verb ἐξῆλθεν in v. 36, in relation to prophecy.)

But there’s also something that to my mind doesn’t quite fit well. Commentators regularly explain the rationale of v. 36 (as a continuation of the line of thought in v. 33) as a biting criticism of the the arrogance of the Corinthian church, vis-à-vis other churches. Horsley asks “[d]o they think they have a unique or special revelation that allows them to behave differently from the other assemblies?” (189); Fee describes “an attempt to get them to see that they are out of step with the other churches” (777); Fitzmyer, “Christian men of Roman Corinth were not the only ones evangelized, and so some respect must be had for Christians in other communities and their customs” (533); Keener, “[t]hey need to recognize that the Spirit has also spoken to the [other] churches (120); and see also Thiselton, 1161.

While a reference to the Corinthian church as a whole would plausibly explain the plural uses of “you” in v. 36, though, the use of the plural doesn’t to my mind actually fit as a reference back to those criticized in v. 33 and before. Ciampa and Rosner follow Fitzmyer, Fee, and others in seeing a reference to the church as a whole, but then also highlight another element: v. 36
would suggest a rebuke of those Corinthians who think they are free to go their own way, independent of any concern about what the rest of the churches do. Paul’s argument is that the Corinthians’ worship should “be in tune with the rest of God’s churches.” We agree with this but also see in the verse a rebuke directed against those prophets who tended to monopolize the church’s time, refusing to be silent long enough for others to have their own opportunities to speak to the congregation. (The First Letter to the Corinthians, page unknown).
But it’s Charles Talbert who, emphasizing the singular/plural disjunction here, most directly expresses this as a criticism of the interpolation view; and he in fact seems to take this as decisive against it:
The difficulty with this reading is that v. 36 is addressed to “you all” (plural), whereas, if a response to vv. 26-33, it ought to be “you” (singular). Paul has been saying in vv. 26-33, “If any one speak in a tongue, let . . . But if there is no one to interpret, let each of them keep silence . . . If a revelation is made to another sitting by, let the first be silent.” That is, he is addressing unbridled individualism and is asking individuals to subordinate their personal expression of spiritual gifts to the corporate good. If he then says indignantly that the word of God has not gone forth from and come to them only, it ought to be addressed to the individuals addressed in vv. 26-33, not the church as a whole or the group of prophets within it. This difficulty is sufficient to tip the scales in favor of taking vv. 34-35 as a Corinthian assertion and v. 36 as an indignant Pauline reply (Reading Corinthians: A Literary and Theological Commentary, 116-17)
A few scholars have suggested that 1 Cor. 14.34-35 originated as a marginal note that was eventually incorporated into the main body of the text (Fitzmyer cites Ellis and Barton for this view). Although this view is premised on the idea of 14.34-35 being an authentic teaching held by Paul himself, and perhaps even added by Paul — e.g. “a parenesis added by Paul . . . which he considered appropriate to his concern about proper order in the Christian community” (Fitzmyer, 530) — maybe it was the other way around. Perhaps Paul knew the teaching in 14.34-35 to be a Corinthian position — whether the current text of 14.34-35 represents an actual quote from their letter, or is simply a paraphrase of the idea — but Paul had simply forgotten to incorporate this as well as a criticism of it into the original completed version of 1 Corinthians, or couldn’t find a good way to do so. Still wanting to incorporate it, however, he found an appropriate place for inserting it where he did, whereby it’d still be refuted by being followed by the preexisting condemnation in 14.36, even if this originally applied only to the previous material before 14.34.

Of course, this also runs against some of what Talbert had suggested earlier re: the likely antecedent of the plurals in 14.36, and which I tentatively supported, too. But if Paul did indeed want to incorporate 14.34-35 along with a condemnation of this into an original version of 1 Cor. 14 — and if in this original version, the antecedents of the plurals in 14.36 did (uncomfortably) refer back to the preceding material about prophetic order — perhaps the space between 14.33 and 14.36 provided Paul the opportunity to kill two birds with one stone. Inserting 14.34-35 here, the awkwardness of 14.36’s syntax would thus be alleviated, now having true plural antecedents: condemning the one who would violate prophetic order, and one who’d object to women speaking in the assemblies.
There are other options for preserving the idea of a Pauline criticism of the Corinthian position while also positing some sort of textual alteration, too.

At least as we know it from other alterations in the manuscript tradition, scribes were just as likely to omit text as they were to add it. Perhaps 1 Cor. 14.34-35 was always present in the Vorlage of 1 Corinthians, and there was originally more to Paul’s refutation of this besides 14.36. However, a later scribe, motivated by whatever it was that motivated them — their knowledge of the tradition in Timothy 2.11-12 and/or their desire to harmonize 1 Corinthians with this, etc. — then removed some of Paul’s more explicit criticism here, leaving only the non-Pauline view in 14.34-35 itself and 14.36.

Incidentally, this is somewhat the converse of what D.W. Odell-Scott has suggested happened for the actual manuscripts which dislocate 14.34-35 from its best-known location:

I suggest the editors of manuscripts D, G and 88, removed verses 34 and 35 from their canonical location at 33/36, and inserted them after verse 40 in order to shelter the silencing and subordination of women from the critique of verse 36 and to positively associate the silencing and subordination of women with Paul’s admonition for decency and order. (“Editorial Dilemma: The Interpolation of 1 Cor 14:34-35 in the Western Manuscripts of D, G and 88”)

Both Gordon D. Fee and Philip B. Payne are notable scholars who believe that 1 Corinthians 14:34-35 is an interpolation. Other scholars who hold this position include Straatman, Fitzer, Barrett, and Ruef. In his article entitled Fuldensis, Sigla For Variants in Vaticanus, and 1 Cor 14.34–5, Payne points out that
“…scribes in that period simply did not take the liberty to rearrange the argument of Scripture in this manner. We do not have even a single parallel example of a scribe rearranging the sequence of an original text of any of the NT letters to make it more logical. Furthermore, even if Bishop Victor felt he had the authority to rearrange the sequence of the text, there is no adequate reason why the text would make more sense reinserted at the end of the chapter.”

In his article MS. 88 as Evidence for a Text Without 1 Cor 14.34–5, Payne claims that:
“The evidence that ms. 88 was copied from a text of 1 Corinthians 14 without vv. 34–5 provides additional external support for the thesis that vv. 34–5 were not in the original text of 1 Corinthians 14.”

Curt Niccum wrote an article entitled The Voice of the Manuscripts on the Silence of Women: The External Evidence for 1 Cor 14.34–5 that claimed the bar-umlauts themselves were added to the text at a later date and are thus not indicative of an interpolation, but rather of a paragraphos (marginal note).

I argue that the transposition of verses 34 and 35 after verse 40 in western manuscripts D, G and 88, does not strongly support the modern interpolation hypothesis which contends that since there are textual deviations, as well as significant inconsistencies if not contradictions between the content of verses 34 and 35 and the rest of First Corinthians, the verses were inserted into the epistle by post-Pauline editors. I review the “egalitarian interpretation” of 1 Cor 14:34-36, and my earlier arguments against the modern interpolation hypothesis. Assuming the egalitarian interpretation, I suggest the editors of manuscripts D, G and 88, removed verses 34 and 35 from their canonical location at 33/36, and inserted them after verse 40 in order to shelter the silencing and subordination of women from the critique of verse 36 and to positively associate the silencing and subordination of women with Paul’s admonition for decency and order. I further argue that the editors assessed that the verses in question were misplaced by an earlier editor for which they offered a corrected edition. I conclude that both the earlier and modern interpolation interpretations of the final verses of the fourteenth chapter of First Corinthians assessed that the canonical text was incoherent. Yet both interpretative projects are unable to resolve the textual incoherence that results with the removal of verses 34 and 35 before the twofold negative rhetorical question of verse 36.