Actually they do, a little bit.
Unlike in the old CRT days though, the burn in doesn’t seem to be permanent, at least not on monitors that I’ve had experience with. Just let them run a screen saver that varies the screen a lot every night for a week or two and the burn-in disappears.
I’m a little fuzzy on the technical details, but as I understand it the liquid crystal basically gets into a state where it doesn’t fully relax back to its idle state. Exercising those pixels eventually gets them functioning properly again.