Brian - flatpanel as in LCD or flatpanel as in flatCRT?
any CRT can have an image burned on it, though it takes a ridiculous amount of time for that to happen with newer CRT's
if it's an LCD, no burn
You don't get burn-in on the LCD, but supposedly you can extend the life of the screen (which is probably fairly long to begin with) by blanking it -- having the computer turn it off, not just all-black -- when you're not using it for extended periods. And you save electricity too.
You won't get burn in, but you can get what's called ghosting, where an image will stay on the screen for several seconds/minutes after what was being displayed is gone -- depending on the quality of the screen. It's not permanent, though.