Will keeping a car mount magnet against my iPhone screen damage it?

Most car mount magnet plates are meant to be placed on a case or directly to the back of the phone. If you are just placing the plate against your screen then it wouldn't cause any problems. Because the plate is metal though I would probably use some sort of screen protector so that the glass doesn't get scratched up.

The metal plate is typically not magnetic - the magnetic part is on the mount. To verify, try to pick up a paperclip or a screw with the metal plate on your case. If it doesn't pick up a screw or paperclip, I wouldn't worry about it. If it is magnetic, then I would be apprehensive about keeping it placed against the screen constantly but someone else here may be able to provide better information on that.