Squeeze a generous dollop of white, non-gel toothpaste in a bowl. Add 1 tsp. of baking soda at a time until a thick paste forms.
Dip the corner of a damp cloth in the paste and begin rubbing the heat marks for several minutes. Continue rubbing until the marks begin to warm.
Wipe the table clean with a clean cloth dampened with water. Repeat the process until you have removed the heat marks off the table.
Pour 1/2 cup of lemon juice into a bowl. Add a 3 to 4 tsp. of cigarette ashes to the juice and stir with a spoon.
Dip a damp sponge in the mixture and begin rubbing the discolorations. Continue rubbing until the heat marks are no longer visible.
Wipe the mixture off the table with damp paper towels.
Rub the heat rings with a fine piece of 000-grade steel wool. Rub with the wood's grain for several passes.
Examine the heat stains and every pass you make with the steel wool. Repeat the process until the discoloration is no longer visible.
Buff wood polish and wax on the table as you normally would.