Use care in selecting your screen protectors. Apparently there are two types of screen protectors, removable or permanent. The difference being that the temporary type use little or no adhesive while the permanent type use an a strong adhesive that forms a permanent bond with the touch screen surface.
Not having done my homework, and wanting to protect my DSi, I purchased the RocketFish DSi starter kit when I purchased my DSi. I immediately installed the screen protectors included with the starter kit without incident and they do work just fine. HOWEVER, if they do wear and need to be replaced I will have a problem. The adhesive used is strong and leaves behind a messy residue on the screen. It is also very easy to destroy the touch screen during the removal process.
(Hint: Once you get a corner loose, pull it back on itself so that you are pulling across the screen rather that pulling up and away from the screen. Pull it off very slowly.)