This is not about speech. In holiness contexts, that which is “profane” is that which is “ordinary” (Blenkinsopp, Isaiah 40–55; vol II, pg 253). To profane something holy (literally “set apart”) is to desecrate it, to defile it—to treat as something not special

For this verse in context, speaking to the Babylonians:
I was angry with my people,
I profaned my heritage;
I gave them into your hand,
you showed them no mercy;
on the aged you made your yoke
exceedingly heavy.
As another part of Second Isaiah (Isaiah 40–55), this pairs insightfully with 43:27–28:
Your first ancestor sinned;
your spokesmen transgressed against me,
so I profaned the princes of the sanctuary;
I delivered Jacob up to ruin,
Israel to reviling.
“Princes of the sanctuary” is a term used in 1 Chronicles 24:5 for the priesthood; however, Blankensopp sees this here as likely referring to “civic and religious leaders” in the vein of Lamentations 2:2 (“he profaned the kingdom and its princes”). He brought them to ruin.

As a whole, we see a strong appeal to Deuteronomic doctrine. Israel and Judah were idolatrous, so in his outrage God actively brought judgment on all the land through their enemies—their misdeeds were a violation of the national agreement with all its stipulations.

It seems, then, that for Judah to be profaned by Yahweh, it is a forceful rejection and removal of their status and station. (And considering that the Judaeans were traumatically reeling from the destruction of the Temple, I also expect the visceral understanding that their all-important sacred space had been literally rendered profane was present as well.)
Commentating on the parallel passage in Lamentations 2:2, Salters supports Blenkinsopp’s analysis, saying:
The meaning … must be associated with Isa 43.28, where Yahweh says ‘I desecrated the princes of the sanctuary’, and with Isa 47.6, where Yahweh speaks of desecrating his inheritance and of handing them over to Babylon. In both of these passages the invasion of Judah by Babylon is interpreted as Yahweh himself polluting or desecrating. Lamentations 2.2 may be the source for the Isaiah passages. … The language used implies the breaking of the special relationship between Yahweh and his people. The verb, which has the basic meaning ‘to untie’, has to do with the profanation of the holy—the treating as profane that which is sacred. The conquest of Judah means that Yahweh no longer regards Zion as a holy entity (cf. Exod 19.6; Deut 14.2).

— R. B. Salters, A Critical & Exegetical Commentary on Lamentations (pg 118)
The idea of Yhwh/God exerting violent retribution on Israel/Judah through foreign armies and the horrors of their invasion is a common motif, especially in prophetic literature —and also a central theme of First Isaiah (chapters 1-39). For quick reference on these specific verses (more context & detail below), see the Jewish Study Bible (2014):
6–7: Babylonia’s hubris. Babylonia never was as strong and self-sufficient as she believed. Her victory resulted only from the LORD’s decision to punish Judah. Babylonia took this commission too far, however, and now she must be punished as well. On the hubris of the foreign conqueror who unknowingly does God’s work, cf. 10.5–15.
And Alter’s translation & commentary:
I was furious with My people / . . . You showed them no mercy. This verse performs a theological balancing act. God was furious with Israel and punished them by allowing the Babylonians to defeat them, but the Babylonians enacted this historical role mercilessly, tormenting Israel with a heavy yoke, treating even the aged savagely. For this, Babylonia will now suffer severe retribution.*
Blenkinsopp:
Isaiah 47 falls into the category of the taunting of the conquered by the victors, […] Ritualized verbal humiliation of a defeated enemy is one of several forms adopted in oracles against a political enemy, a genre well represented in the prophetic books (especially Isa 13-23; Jer 46-51; Ezek 25- 32). This type of poem seems to have developed out of brief oracular sayings or imprecations uttered by a seer before a campaign (e.g. Balaam in Num 22-24) accompanied by homeopathic ritual acts—such as, for example, the smashing of a pot (cf. the so-called Execration Texts from the Egyptian Middle Kingdorn, ANET 328-29). In its present form, this particular example is couched in oratio recta by Yahveh, though the attribution to Yahveh occurs only in the textually insecure beginning of v 4 (see Notes on the text) and v 6, which is often, and perhaps correctly, read as a later interpolation. […]

Though the language of the poem has several peculiar literary features and a high percentage of rare vocabulary words, it is well enough integrated into chs. 40-55 (pace Merendino 1981, 489, 495, who regards it as an interpolation): the fall of Babylon may have been hinted at right at the beginning of Second Isaiah, in the allusion to human impermanence (40:6-8 cf. 51:12); the capture of the city has been described as part of the mission confided to Cyrus (43:14; 45:2-3; 46:11); and the omens and divination characteristic of Baby- Ionian religion and the cult of Bel and Nebo (Marduk and Nabu) have been dismissed as futile (44:25; 46:1-2, 5-7).
More significantly, the female persona of Jerusalem-Zion is presented as a mirror image of the dishonored queen of Babylon: whereas the latter sits in the dust, the woman Zion is told to get up off the ground (52:2); and whereas the Babylonian is forced to expose herself, Zion is told to put on beautiful clothes (52:1). The one is widowed and bereaved of children, the other will no longer be bereaved and will have numerous children (49:20-21; 54:1). The one is shamed, the other will no longer experience shame (54:4).
The issue of chronological order and dependence is complicated by the tendency of this kind of diatribe to use stereotypical phrases and adopt the same traditional language of abuse and denigration. This type of saying can very easily also be “recycled” in accordance with the changing international scene. Nahum, for example, is directed against the Assyrian capital Nineveh described as a lascivious woman (zona) addicted to divination (ba’älat kësâpîm 3:4), whose fate is to be stripped and displayed in public (3:5-6)—not very different, therefore, from the woman Babylon of Isaiah 47. But the name of the city, which occurs only twice in Nahum (2:8; 3:7), could easily have been removed and the polemic redirected, practically unchanged, against Babylon.

Feminine personification was traditional for cities (Babylon, Jerusalem, Nineveh; Sidon in Isa 23:12) and peoples (Egypt, Jer 46:11; Judah, Lam 1:15; Israel, Amos 5:2; Jer 18:13; 31:14, 21). In the case of hostile cities such as Babylon, an unfortunate corollary was the unleashing of violent and at times pornographic imagery focusing on the female body. That several of the commands addressed to the woman Babylon in the first strophe (1-4) are mutually incompatible (sitting on the ground, undressing, grinding corn, wading through rivers) suggests that the author of the poem simply assembled a congeries of images to create an impression of degradation and shame.