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).