Take a sharp scraper and gently remove all the finish over the plug/bung. Then take a small screwdriver or ice pick and push it into the bung (a hammer can be helpful). Chip/pry out small pieces of the bung without damaging the area surrounding the circumference of the bung. Patience.
Hopefully the bung was held in place with varnish or or the like. If glue or epoxy, damage getting it out may be inevitable.
Usually, there is a screw under the bung.You may have to use a screwdriver/ice pick and hammer, gently, to clear out whatever is in the screw slot before you can attempt to remove the screw.
Repeat for all bungs.
Upon reinstalling the trim, think of the next poor soul and do not use glue/epoxy to reset new bungs. Use polyurethane or varnish.
Then use a chisel, carefully, to trim the new bungs down almost, but now quite, to level. Then gently sandpaper the bung flush. Then sand all the trim and recoat with whatever is whatever is compatible with the previous finish.