They're never described as talking in any of Tolkien's writings that I could find. However, they can certainly make noises. Some actions can be interpreted as being spoken and some events would be unlikely to have happened without speech, but most of it is circumstantial.
That being said, these are books born out of a love of language and speech. You'd think Tolkien, a philologist, would have mentioned that a whole group of sentient beings were unable to speak or (worse!) had not developed a form of language.
So there's nothing really concrete, but here's what I found anyway.
Gothmog
Gothmog started as Kosomot, Son of Melko. He was "the 'marshal' of the hosts of Melko". A marshal is usually one of the highest ranks in a military, but it's not a very precise term as its definition has varied by period and location. It would however be difficult to command a whole army without talking.
In an early name-list, the element -mog is said to mean "voice", as in "Voice of Goth (Morgoth)". This would later change to "strife and hatred", but there's also this footnote:
Nothing is said in any text to suggest that Gothmog played such a role in relation to Morgoth as the interpretation 'Voice of Goth' implies, but nor is anything said to contradict it, and he was from the beginning an important figure in the evil realm and in especial relation to Melko. There is perhaps a reminiscence of 'the Voice of Morgoth' in the Mouth of Sauron', the Black Numenorean who was the Lieutenant of Barad-dur.2 p.67
There are also several passages in the early drafts of The Fall of Gondolin that mention Gothmog bidding or ordering people:
But now Gothmog [...] gathered all his things of iron that could coil themselves around and above all obstacles before them. These he bade pile themselves before the northern gate[...]2 p.176
Then Gothmog Lord of Balrogs gathered all his demons that were about the city and ordered them thus[...]2 p.179
Finally, Gothmog is said to have "mocked" Húrin (mentionned by Ben Osborne above):
Gothmog bound him and dragged him to Angband with mockery.S p.231
Others
Balrogs are said to laugh at Húrin in early drafts of The Lay of the Children of Húrin:
and the Balrogs about him brazen-handed
with flails of flame and forgéd iron
there laughed as they looked on his lonely woe
An early draft of The Lord of the Ring has the Balrog hiss just before falling from the bridge:
With a gasping hiss the Balrog sprang up; it seemed to be [?half blind], but it came on and grasped at the wizard.7 p.198
The final text has the Balrog scream:
With a terrible cry the Balrog fell forward[...]I p.434
The Balrog who kills Glorfindel in Gondolin also shrieks:
Then Glorfindel's left hand sought a dirk, and this he thrust up that it pierced the Balrog's belly nigh his own face (for that demon was double his stature); and it shrieked, and fell backwards from the rock[...]2 p.194
References starting with a number refer to a History of Middle-Earth volume, roman numerals to a Lord of the Ring part and 'S' to The Silmarillion. Page numbers from the Harper Collins editions.