| If there is a finish on you dresser - i.e. lacquer, polyurethane, etc. and simple furniture polish applied with a cloth does not remove the circles, then I would try some 0000 steel wool with furniture polish or paint thinner and rub very lightly.
Another type of liquid product you might want to try that usually comes in small cans are those that remove things like dried paint or gum. I think that one of them is called "Goo Gone."
If you have no finish on your dresser, than you will probably need to use 150 to 240 grit abrasive paper. | 
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