Screen damage is a tricky thing. Apple does replace screens, but not very often. It could be a heat issue if you charge in the case (just about everybody does) but Apple will look for just about any reason for potential damage caused by the user. If you hve any evidence of damage they will not replace it either. My best advice is to take it to an apple store if you can and see if they will replace it for you or go here
http://depot.info.apple.com/ipod/
You never know, maybe they will.
but if not you can replace the screen yourself for well under 20 dollars and it is not that hard of a thing to do if you have any mech skill. But only do that if it is off warranty or they will not repair it for you. Go under ebay and search for the screen, they come with instructions and take apart tools. see this take apart guide.
http://www.ipodmods.com/guides/Nano/nanodisassemblyguide.htm