Your bathtub constantly has water flowing through it, putting any metal in the tub at risk or rusting or corroding over time. When nobody has a reason to remove the overflow cover from the bathtub for years, the screws that anchor the cover in place may rust to the point you cannot turn them with a screwdriver.
Before you try to force the screws out and potentially strip the heads or shear the screws, spray some penetrating lubricant into the screw holes and let the lubricant soak in for an hour. Depending on the severity of the rust, you may need to apply the penetrating lubricant more than once, stopping after you back the screws out slightly so you can spray into the threads more easily.
Using the right size screwdriver will help make the job easier, but the screwdriver's head can still shift around in the screw heads, making the job unnecessarily difficult. By packing an abrasive household cleaner or even sand into the screw heads, the tip of the screwdriver will have some grit to keep it in place so you can turn the screws out of the trip plate.
When all else fails, you will need to turn to vise grips to turn the screws out of the holes so you can remove the trip plate. You first must use a close-quarters hacksaw to trim the screws' heads off, leaving just the posts protruding from the screw holes. Use your vise grips to grab each post securely, and then begin the slow process of turning the posts counterclockwise until the screws unthread from the trip plate.
Bathtub trip lever cover screws will come into contact with water on a regular basis, leaking to corrosion if the screws do not have protection. You can help prevent rust buildup by regularly removing the screws from the trip plate and brushing off any rust or corrosion using a wire brush. Removing the screws on a regular basis requires you to constantly remember to do so. Instead, before you reinstall the screws, brush off any corrosion with a wire brush, and then spray the screws with a clear acrylic spray. Allow the screws to dry, and apply one more coat. The acrylic spray will help protect against rust and corrosion in the future.