I struggled with this, too. After following several dead ends (from Google searches, e.g., suggesting deleting photo caches, which had no effect on the warning messages for me), and not knowing of @hawkeye's useful information, I noticed that there was a small warning symbol (triangle with an exclamation mark in it) in iTunes, between my device's name and the device eject symbol, at the top of the device panel on the left. When I clicked on the warning symbol, a message popped up describing what the problem was; see the image. (In my case, I didn't have enough free space to replace a particularly large app.) Apple sure could make this more obvious!
By the way, you can get an unsynced item warning on the device when there isn't a problem, due to stale data in iTunes. You can eliminate the device warning by resetting dialog warnings and the iTunes Store cache in the Advanced section of iTunes preferences; there are two buttons there you can push. This solved a mysterious sync warning problem for me in the past.