I realize this thread is old but it's relevant to my current situation, may be to others as well...
A few of my neons (cardinals) have this mouth bubble as well. One of them had the mouth bubble grow so large that he was unable to eat. I knew that if something didn't change that he would die shortly, so after quarantining and treating (no change) I actually made the decision to cut it off.
Yes, I know. I had no idea what to do but I knew that either he was going to die from starvation or die from his bubble being cut off, so I had nothing to lose.
In the end? I held him at the top of the tank and snipped it off. It was cut off with a pair of tiny sewing scissors and he didn't seem to feel a thing, he was just annoyed that I was holding him (while in the water, of course). I examined the "bubble" and it was indeed NOT a bubble at all but a solid piece of flesh. I am still clueless as to what it is/was, but I can happily report that the little guy is healthy and happy, can eat just fine now. There are still signs of his "bubble" - his mouth can't close all the way anymore - but at least he can live normally.
All that being said, I do not recommend snipping off these bubbles UNLESS it is a matter of life and death. But if you must, be assured that the fish can do just fine after the bubble is removed. AND it may come back, just an FYI.