How can I remove a plastic cupholder glued to the wood frame of a large mirror? I don't want to damage the frame or mirror
Get a 3" wide plastic drywall knife and cut off the handle. Wrap masking tape around the cut end, and you can use it like a saw. A second plastic drywall knife 1" wide can be used to pry away the cupholder as you cut.
After the cupholder is gone use the "saw" to cut away the glue, just like slicing bread. If the wood is painted you can remove the glue and repaint. If the wood is plastic coated you could test a solvent like alcohol or artist thinners. If the wood is varnished, look for a goo-gone product safe for varnish. Coconut oil dissolves many things, but test it first on a hidden edge.