Hot pans, pizza boxes and coffee cups can cause heat damage to finishes that resembles the off-white look of water rings. Fixing the damage can be difficult. Your success is going to depend a lot on the finish.
There are basically two ways to clear up the finish: redissolve it so it dries back clear or abrade the finish to below the damage.
Shellac and lacquer are the only finishes you can redissolve. Shellac was popular until the 1920s. Lacquer has been popular since. You can test for the type of finish by dabbing a little denatured alcohol (for shellac) or lacquer thinner (for lacquer) onto an inconspicuous spot and see if the finish gets sticky.
To redissolve the finish, spray some alcohol or lacquer retarder onto it. Spray very light, mist coats. Don’t wet the surface or you risk the finish separating or bunching up into “fish eyes.”
This fix won’t work on varnish, polyurethane or on modern high-performance finishes, which are being used more and more on tables, especially those made overseas. And quite frankly, I’ve never tried repairing this type damage on a water-based finish because I’ve never seen a heat mark on this finish.
Abrading will work on all finishes, but not well on high-performance types, such as conversion varnish. These finishes are so hard that it’s very difficult to scratch them. If the alternative is to refinish anyway, you could always try abrading through the damage (which is almost always very near the surface) with coarser sandpaper. But you’ll then have to rub the finish to an even sheen with finer abrasives, which may not be possible because of the hardness of the finish.
If you are able to remove the damage by rubbing with an abrasive, you’ll then need to rub that area to the sheen of the rest or the surface, or rub out the entire surface to an even sheen. Either of these can be done on lacquer, shellac, varnish or water-based finishes.
Alternatively, you may be able to match the sheen closely enough by spraying light coats of finish (gloss, satin, flat) onto the abraded area using a spray gun or aerosol