Try MENU>SETTINGS>PRIVACY the uncheck display message text.
your phone screen shouldn't stay lit up anyways, but this is the only simple solution i can think of.
The other option is an app you downloaded recently screwed something up; for example: I downloaded the swype beta and loved it, they came out with an update and after I installed it my screen would never time out by it self, it would stay lit for hours unless i pressed the power button. With an uninstall of swype, it worked perfect again. I have tried reinstalling swype a few times since then because I liked the app but to this day if it is installed my screen will not timeout.
My guess is its some error in the permissions, but I am the only one who was having problems with it from the sounds of it. I would try to uninstall the last app or two you downloaded before this started happening.