If it seems to be temperature related, then you don't have something burnt, you likely have a broken solder joint or trace somewhere. Can't really give you a specific point to look for, since I don't know what connections on the display do what.
My best guess, since it works when it's cold, and metal contracts when it's cold, would be to say maybe one of the pins is getting pulled inward enough to make solid contact while cold. And expands a tad when it warms up to break contact. You could try touching up each of the solder joints with your iron, to reflow the solder and seal up any breaks. No guarantee that's the issue though.