It depends on the screen, but in plasma and CRT screens, the phosphor (light-emitting substance) is slowly degraded by electron bombardement, in OLED displays the current damages the organic molecules, and in LCDs a constant activation makes the liquid crystals worse at returning to the deactivated state. On a screen which displays many different things, this degradation affects the whole screen pretty evenly, but if it displays the same image for months or years, the damage forms a visible pattern.
Edit: regular LEDs also dim with use, so those displays can be burnt in as well. When fuel got expensive enough for the price to start with a 2 instead of a 1 in my country, big gas station seven segment LED displays had the previously unused segments in the number 2 shining much brighter than the rest.
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