Professor Olsen does an excellent deconstruction of this in his original "Tolkien Professor" podcasts, but I'll cover the basics really quickly.
In the text, it specifically says "the shadow about it reached out like two vast wings." LOTR Bk2Ch5. There is, of course, other, more circumstantial evidence against the Balrogs having functional wings.
Firstly, there have been three balrogs killed during the history of Middle-Earth. Two, including Gothmog, were killed during the Fall of Gondolin. The third dies in the above-quoted from fight with Gandalf. All three of them are killed falling from a great height, which suggests that they were not able to overcome gravity in any way (such as flight).
Secondly, Melkor was unable to accost Arien in her daily flights over Middle Earth because he had no servants capable of doing so. Had the Balrogs the ability to fly, they certainly would have been able to do this.
Now, there are passages mentioning the Balrogs in "flight" on several occasions during the Silmarillion, but if the use of that word is evidence for winged Balrogs, then there are many instances in which elves, men, and orcs should also have wings.
What was decided some time ago for this project is that the Balrogs arrive on Arda with Melkor, winged and beautiful. They cast down the Lamps of Almaren (influenced by Melkor), and are caught in the Lamps' ruin. The injuries they receive are what turn them into the "demons of shadow and flame" that we all know and love. They have only the "shadow" of their wings left as a bitter reminder of their fall.