I'd be inclined to try a heavy-duty (automobile?) polish or even rubbing compound, to try and rub out whatever is there using an old towel. Sounds like it could be either something in the bottom of the paper plate that the heat caused to transfer to the table's finish, or possibly even something in the finish itself, made cloudy by excessive heat. If those don't work, the next step would be a soft steel wool, just enough to gently abrade the finish to take off the "whiteness." You may have to do some touch-up in the rubbed area using a clear preservative, to match the table's original finish.
Can't blame you for not wanting to refinish the entire table, although that's still an option if all else fails.