This question arises due to a passing comment in one of Tolkien's letters;
"20. The Balrog never speaks or makes any vocal sound at all. Above all he does not laugh or sneer. .... Z may think that he knows more about Balrogs than I do, but he cannot expect me to agree with him."
Letters #210
This could be taken to indicate that what Tolkien knew about Balrogs that 'Z', script writer Morton Zimmerman, did not was that they COULD NOT speak or make vocal sounds. The alternative view is that Tolkien meant only the Balrog of Moria DID NOT do so. It is noteworthy that Tolkien was himself incorrect in at least that the Moria Balrog DID make a vocal sound;
"With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished."
FotR, The Bridge of Khazad-dum
In an earlier draft of this same passage the Balrog makes two sounds;
"With a gasping hiss the Balrog sprang up ... With a terrible cry the troll fell after it, and the Balrog [?tumbled] sideways with a yell and fell into the chasm."
ToI, The Bridge
Both of the other two detailed accounts of a Balrog's death also have it crying out;
"Then leapt Ecthelion lord of the Fountain, fairest of the Noldoli, full at Gothmog even as he raised his whip, and his helm that had a spike upon it he drave into that evil breast, and he twined his legs about his foeman's thighs; and the Balrog yelled and fell forward..."
"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..."
BoLT2, The Fall of Gondolin
We even see Balrogs laughing in an early text;
"Then the Lord of Hell lying-hearted
to where Hurin hung hastened swiftly,
and the Balrogs about him brazen-handed
with flails of flame and forged iron
there laughed as they looked on his lonely woe;"
LoB, The Lay of the Children of Hurin (second version)
We are thus left with the possibility that Tolkien might have decided to change FotR to remove the Balrog's cry, that it could make sounds but not form actual words, or that the Balrog could have spoken but did not. On the matter of speech it has been suggested that the 'counter-spell' scene might be relevant;
"'What it was I cannot guess, but I have never felt such a challenge. The counter-spell was terrible. It nearly broke me. For an instant the door left my control and began to open! I had to speak a word of Command."
FotR, The Bridge of Khazad-dum
The argument here is that we frequently see 'spells' in Middle Earth being spoken aloud; Gandalf himself here speaks a 'word of Command'. If we suppose this to be a universal requirement then the Balrog's counter-spell would have to be spoken. However, this is in the end only supposition based on outside evidence. Another commonly raised point is that the Balrogs commanded forces and must therefor have been able to speak to give orders. Indeed, there are even references to Balrogs giving orders;
"But now Gothmog lord of Balrogs, captain of the hosts of Melko, took counsel and 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..."
"Then Gothmog Lord of Balrogs gathered all his demons that were about the city and ordered them thus..."
BoLT2, The Fall of Gondolin
While these seem clearly to show Gothmog giving orders it can be argued that these very early texts could have been rejected or even that Gothmog issued his orders telepathically or by directly influencing the wills of those under his command. We see examples of such things by Sauron, Elves and even Humans elsewhere.
Finally, in one passage Gothmog's name is translated as 'Voice of the Master', which might imply that Gothmog himself must have had a voice to thus speak for Morgoth;
"Gothmog '= Voice of Goth (Morgoth), an Orc-name.' Morgoth is explained at its place in the list as 'formed from his Orc-name Goth 'Lord or Master'..."
LROW, Appendix II - The List of Names
However, this is again very early text and that etymology was later changed with Gothmog coming to mean 'Dread Tyrant'.
In all, there is quite a bit of evidence for Balrogs being able to make sound and some few bits suggesting that they could speak, but of them all only the single cry as the Moria Balrog fell was ever published by Tolkien... and he subsequently contradicted that one.
