As far as burn in - Yes, some LCD monitors can suffer from burn-in, but it takes a lot of time on a static image to do it. For instance, I bought 10 monitors for a control system and they were kept on the same screens for a number of years before some started to fail. When replaced, These monitors had burnt-in images of the basic control screens. These were 4x3 florescent backlit flat panel LCD monitors.
When upgrading and expanding the system, I bought 16 newer LED backlit 16x9 monitors. They've been stuck on the same screens for about 2 years and none are showing signs of burn-in. This is an extreme environment for the monitors, so I'd say that the newer LEDs are immune.
TL

R - Older LCD monitors can suffer from burn-in, but I haven't experienced it with newer LED monitors.