We had some success with removing white rings caused by hot plates when we had a restaurant some years back by using metal polish as soon as the tablecloths were taken off at night. The rings are caused by the polish on the table being cooked. Sometimes the ingredients in the metal polish would sink in and make the ring appear darker. Sometimes if the ring was just on the surface the metal polish would remove most of it ready for re-polishing. We used silver polish (The stuff you have to shake up till it turns pinky-white) and a soft cloth and circular strong movements. The jewelers polish (the pink stuff) was finely abrasive but did not scratch the surface. It did not work on every stain but the small lighter ones were removed and if the mark was still warm from the plate it would completely remove it. It was no good on water marks though.
Fay