I usually pound a putty knife / paint scraper under the edges of the plate to raise the nails far enough to grab with vice grips so you can twist and pull them. Often the metal plate will get bent up a little, but it can be pounded flat again by putting it face down on an anvil. You may need a lot of prying/force to get the nails to move.

if you're refinishing the side rails, you'll need to remove the plates from okoranges rail to refinish that as well.

ya got nothing to lose ... stick the prying device under the formica and lever up the plates a little. You'll want to start with something thin and heating the formica will probably help detach it from the glue.

after removing the formica, you'd need to sand your rail to remove the glue anyway, so scratching the wood a little won't matter.

if nothing else because OKO's plates are in better shape, remove your coin entry plate. Those people sometimes need.