Place a cloth over the rings. An iron set to medium heat and iron over the cloth. May take a few passes.
You can wipe with tung oil. This will help moisturize the wood and bring back an even finish. :)
Try rubbing mayonnaise on it
Try making a mix of equal parts mayonnaise and ashes. Apply it to the ring mark. Let it sit for 10 minutes then rub in & remove. Good Luck!