1 Kyros: Where's Lambert?
1 {in the foreground, a rat runs across the dungeon corridor}
2 Balrog: RAAARRRHH!! {everyone turns to look at the Balrog lurking behind Kyros}
3 Alvissa: A Balrog! Run!!! {everyone but Kyros runs}
3 {a white cat chases the rat across the dungeon corridor}
4 Kyros: Nobody wants to answer my question?

I'm not sure if Balrogs can talk or not. But I'm guessing they can at least roar terrifyingly.

[30 seconds of Googling later...]

In fact it's not clear to a lot of people whether Balrogs can speak or not.

Just don't mention the wings...

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2021-02-13 Rerun commentary: A more recent discussion of the topic of whether Balrogs can speak or not also fails to reach any consensus.

Quotes, in favour:

They almost definitely could talk, probably Melkian.

Balrogs are corrupted Maiar. We know that other Maiar can speak (e.g. Gandalf, Sauron). If balrogs can't speak, it would mean they lost that ability in their transformation, which there would need to be evidence of.

Durin's Bane not speaking can definitely be taken as circumstantial evidence, but it needs to be weighed against the other things we know about Balrogs.

• Gothmog [Ed: the Lord of Balrogs, not the Lieutenant of Morgul portrayed as an orc in the movies] is called "high-captain of Angband". Captains give orders. Yes, it's possible that Gothmog communicated telepathically, but since there was a spoken language used by Morgoth's servants (Melkian), it's simpler to assume that Gothmog communicated via speech.
• When Gothmog captured Hurin, the text [of The Silmarillion] says he "bound him and dragged him to Angband with mockery." So he was mocking Hurin. The most straightforward interpretation is that it was verbal.

Given those things, I don't see any compelling reason to believe balrogs can't speak.

Neutral:

We don't know.

There are no instances in any of Tolkien's writings of a Balrog speaking, so it's not clear whether they could, or what language they would have spoken. They were, however, definitely able to communicate verbally in some fashion; at the very least, they were capable of making sounds, as Durin's Bane does in the text:

Gandalf lifted his staff, and crying aloud he smote the bridge before him. The staff broke asunder and fell from his hand. A blinding sheet of white flame sprang up. The bridge cracked. Right at the Balrog's feet it broke, and the stone upon which it stood crashed into the gulf, while the rest remained, poised, quivering like a tongue of rock thrust out into emptiness.

With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished.

Fellowship of the Ring Book II Chapter 5: "The Bridge of Khazad-dûm"

It seems somewhat unlikely to me that Maiar spirits, even ones locked in an incarnate form, would be completely unable to speak, especially given their position within Morgoth's forces; even Huan was capable of it, though only on special occasions.

That being said, it's possible that they didn't need the ability to speak; there's evidence that the Ainur were able to communicate telepathically with other creatures. Of course, whether the Balrogs would have retained this ability when locked in their incarnate forms is a question we cannot answer.

And negative:

No: they don't talk at all.

From Tolkien's Letters, Letter #210:

The Balrog never speaks or makes any vocal sound at all. Above all he does not laugh or sneer. ... Z[immerman] may think that he knows more about Balrogs than I do, but he cannot expect me to agree with him.

One might point out that the last comment by Tolkien—"The Balrog never speaks or makes any vocal sound at all."—directly contradicts the text of The Fellowship of the Ring itself, which states that, "With a terrible cry the Balrog fell forward." - and quite possibly the text of The Silmarillion stating that Gothmog mocked Hurin.

Next question: Do Balrogs put cream on their scones first, or jam?