If the stains are “whitish” in color, Maggie tells me that you can gently rub the stain /w a little rubbing alcohol ( isopropyl, denatured, or absolute alcohol), then blow dry it (hair dryer). The stain should disappear.
If the stains are “dark or black”, they have already penetrated the finish, and you'll have to remove the finish to get to them. After stripping the finish (fine steel
wool, or ‘Scotch’ pad?), you can bleach such stains out with oxalic acid (1 tablespoon of oxalic acid in a cup water), or hydrogen peroxide. After bleaching the stains, of course, you would have to apply a new surface finish