How do I remove this heat stain from my table?

Try putting down a towel and then running a hot iron over it. You may have to make a few passes.

This link may be able to help you: http://www.gildandco.com/remove-white-heat-stains-wood-furniture/

Sandie most of the time it is from old wax on top. I purchased a kit from formbys. Called facelift kit. Works awesome. Some folks say never change surface it will devalue it. When it looks like this time to change the finish.

Sandie: What about refinishing it in a shabby chic way?

https://homeguides.sfgate.com/shabby-chic-wood-table-55434.html

The "heat" stain is a common problem with lacquer finish. It is caused by trapped moisture as the surface dried before the water could evaporate. As a cabinet maker we spray a lot of lacquer finish. You can spray the area with a light coat of lacquer thinner and let dry slowly on a dry day (low humidity). or lightly sand the whole surface and spray a light coat of lacquer over the top. The white should disappear in front of your eyes.

Natalie has got it. That is what I do. That is a very bad stain. Place a clothe over it. Iron set to medium and iron over the cloth. You will need to make a few passes. You can also try to set the iron to high.

sand it down and refinish it

Yes, I'm afraid I would decide to sand it down to remove the digs etc. and then refinish it. That would give the best outcome. That said if the digs etc. are part of it's history, you could say the damp stain is too, but to help darken it a bit, rub metsl polish into it or Ash, or both.