Update: The bug was fixed and today, 28th of July 2012, YouTube's playlist autoplay is working again. I can't help think maybe it was an experiment to see how much of the traffic was playlists. Whether intentional or not, the statistical information this incident created should be very interesting.
Today I opened a YouTube page, loaded one of my music playlists and started listening to the music in the background. After the first song played, the next started, only with the Autoplay button off, so when it ended, the entire playlist play stopped.
No, this is not you, it's not from your computer settings, Flash updates or anything like that, it is a YouTube issue that keeps turning off the autoplay feature. I don't know if it is an attempt to force people to look at the pages with ads on them or simply an accident - my guess is that it is a bug - but this annoying behaviour started today. The funny thing is that if you press the "Next video" button, the Autoplay button gets back to on, another argument for this being a bug.
I've found a Google topic about this problem, you can find it here, also a Yahoo Answers question, that you can find here.
I don't have a solution for this, but I will update this post if I find it.