First try to remove them by taking a steam iron and heating
the white spot through a damp cloth. The spots are due to water
vapor that gets trapped in the surface of the finish and the heat can force the water
out. For some reason, the steam in the iron helps to prevent the wood from getting scorched. If that doesnt work you will have to gently abrade the surface without damaging the wood. You can try either 0000 steel
wool with a lubricant like paste wax or lemon oil
. I found this quote on a Bob Vila web site which will accomplish the same thing:
If the wood has a good finish (don't try on bare wood), mix equal parts
of baking soda and regular white, nongel 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).