Yes as Kappy said above, if you put your whole computer to sleep while you're downloading data (or say you want to step away from the computer for awhile and stay signed in on IM or something), it will stop. Though if you put the display to sleep/goes to sleep on it's own, you'll be good.
Wether I put it to sleep, or let the display go to sleep by itself, browsers/photoshop/whatever is still generally up. I haven't had any image persistance problems, and I've had this thing for 5 1/2 years now (it's still possible though with any LCD monitor though)