Those white rings are usually moisture/heat trapped in the table’s finish (often lacquer or polyurethane), not damage to the wood itself. Start with the gentlest method first:
Clean the surface
Wipe with a damp microfiber cloth and a little dish soap.
Dry completely.
Use gentle heat (works surprisingly often)
Put a clean cotton T-shirt or towel over the marks.
Use a hair dryer on medium heat or an iron on low/no steam.
Keep it moving for 10–20 seconds at a time.
Check frequently. The white haze often disappears as the moisture leaves the finish.
Dishware sets
If marks remain: petroleum jelly or mayonnaise
Rub a thin layer onto the rings.
Leave for several hours or overnight.
Wipe off and buff.
For stubborn cloudiness
Try a small amount of white toothpaste (not gel) mixed with baking soda.
Rub lightly with a soft cloth following the grain.
Wipe clean immediately and polish.
Finish with furniture polish or wax
This evens out the sheen and protects the surface.