I'm surprised that this question hasn't been asked before (here or on Physics), to the best of my knowledge. It's one that I might have asked when I was a bit younger, and one that I think other people will ask.
Anyway, it's clear that Saturn's rings won't form a moon, and the same is likely to be true for other ring systems. However, I'm guessing that they won't last forever (it's just a guess).
How long do planetary rings in general last? What mechanisms could cause them to dissipate/fall apart/end? I'm guessing the Poynting-Robertson effect could come into play, but I'm not sure.
And for anyone curious, yes, I checked just for the fun of it, and Yahoo Answers had a bunch of really, really bad, unsourced and most likely inaccurate answers (given that there was no consensus), ranging from '3 million years' to '13-18 billion years' to 'forever'.