This is likely a lacquer or ployurethane finish.I would try to find a
solvent that will remove the foam and would not scrape. If it's poly,
any solvenet is pretty much safe. If it's lacquer you need to be more
careful.

Here is what I would do:

On an inconspicious place where you have the same finish try rubbing it
with lacquer thinner. If it doesn't wipe off the finish, then try using
the lacquer thinner to remove the foam backing.

If that doesn't remove the foam, try acetone and ammonia in the same
fashion.

If you find that it is lacquer, ie the lacquer thinner does begin to
make the finish tacky and ultimately wipes it off, and the acetone or
ammonia don't remove the foam, then you could use lacquer thinner and
get the foam off without scarping up the table and just recoat and
smooth out with some more lacquer and sanding, etc.