Soak a cotton ball in rubbing alcohol, then wipe the soaked cotton over the entire surface of the glass coaster to saturate the paint. Allow the rubbing alcohol to sit for one minute to remove any surface varnish from the coaster.
Soak a cotton ball in acetone, then wipe the soaked cotton ball over the glass coaster to saturate the paint with the solvent for one minute. This will loosen the baked-on paint.
Scrape the glass coaster with a razor blade to cleanly remove the softened paint. Continue scraping the coaster until all of the paint is removed, or until you have removed as much paint as you can.
Apply more acetone to particularly stubborn baked-on paint, and allow it to sit for two to three minutes before scraping the paint off with the razor blade.