Squeeze non-gel toothpaste into small bowl. Mix enough baking soda into the toothpaste to create a paste.
Rub the paste into the heat rings with a damp cloth. Rub until the paste begins to warm.
Wipe the paste off the wood table with a damp cloth. Repeat the process as needed until you have removed all signs of the ring.
Rub the heat rings gently with a piece of fine, 000-grade steel wool. Rub with the wood table's grain.
Continue rubbing for a few passes. Examine the heat rings after every other pass. Repeat the process until the heat rings are no longer visible.
Apply wood polish and wax to the table as you normally would.
Mix together cigarette ashes and lemon juice in a small bowl.
Dip the corner of a cloth in the mixture. Rub the heat rings with the mixture.
Continue rubbing and apply the mixture until the heat rings are no longer visible.
Wipe the wood table clean with a damp cloth. Dry with a towel.