There's no way an overheating GPU will destroy a display. Unless the GPU catches fire and the fire spread across the display cable... You're more likely to be hit by lightning seven times than that happening. Then again, that has happened to one person, so...?
Worst case would be a (-n unnoticeable) shortened life of the GPU, but a GPU has built-in throttling to prevent them from getting too hot.
There are very few current high-end cards able to game on a 4k display without reducing game detail. So a four generations old low-end GPU would obviously not be able to do much other than play Mine Sweeper. 4k video at 60 FPS might also cause constant video slow-downs in intense scenes with that amount of graphics memory. I'm not exactly sure of where the limit is, but I think the low bar for fluid 4k 60 FPS video might be around a 4GB 750Ti depending on your computer's overall performance. Basic Windows usage and web browsing will probably feel exactly the same way it does on your current display.