If the wood has a good finish (don't try on
bare wood), mix equal parts of baking soda and regular white, non-gel
toothpaste. Lightly dampen corner of a clean, soft white cloth with water
and dip into the paste. With circular motion gently buff the marks for a few
minutes. Wipe area clean, and buff to a shine. Follow with furniture polish.
(If rings remain after buffing five minutes or so, they may have penetrated
the wood; you might have to refinish the piece). If that doesn't work, dip a

cloth in vegetable oil, then in cigarette ashes, then rub it over the mark.
Another method is to rub real mayonnaise onto the stain, allow to sit
overnight, then wipe with a dry towel.