Are you accessing videos through app or web browser? If you are using browser are you using stock browser?
The app. I think I have also tried on the browser and that didn't work either if I recall properly. I use the stock browser, which has a tendency to crash on me. I tried another browser from the market and that crashed on me too. (I wasn't trying to do YouTube when it crashed, just another special thing about the droid). I also tried to play a movie someone sent in an attachment and that didn't want to play either.
I tried the browser YouTube (nonmobile) on my droid last night and that worked. I think Droid just doesn't like playing videos not on the internet.
