How do I remove these spots on my oak table?

Those are called ‘medullary rays’ and are part of the wood. They are highly prized and are found in oak. Although they are present in red oak they are much more pronounced in white oak. From your picture it looks like quarter sawn white oak and the rays cannot be removed.

I use a sponge on mine, as maple, it works fine. However it may be gummy and need heat to loosen it or be a Permanent mark from heat. How much have you placed hot drinks on the table? Is the question you need to ask, as that is really the only way besides outright defacing to get close to Permanent Mark in a Quality Coat of Varnish that is not a Veneer.

I know it is hard to suggest that these give your table Character like Wrinkles on an old man, so I will not detail that here.

Steamy compresses usually assist.

Try boiling a few dishrags in water and laying those with tongs on top of the rings. A Steaming Rag on top of each usually helps with any impermanent marks on Real Wood that is suffering a Chemical reaction with a Polish like Formby's Polish and is not a Veneer but instead solid wood.

If it is veneer, (yours aint) then you may want to use an Iron with a steamy flimsy rag rather than Steamy Compresses with thick rags only.

Once you fix them, be sure to use coasters, trivits, placemats, runners or tablecloths.

Thank you for your comments and suggestions.

As a former antique dealer specializing in antique furniture, I would suggest a product to use that works like a charm to remove those white heat and water marks and restores the finish to like new. I highly recommend it and it comes in a myriad of wood colors so it is easy to match your finish. No sanding, wipe on and wipe off. I have provided a link to their website, and there are several videos available. Here is the link:

http://www.howardproducts.com/product/restor-a-finish/

It is sold at Home Depot and Lowes as well as online. Best wishes.

I'll check it out next time I’m in town, thank you!

In the past I have placed a blotter over the spot and placed an iron (not very hot) on the blotter. Don't leave the iron more than a few seconds. It has worked for me.

Lots of good solutions—thank you all!

Cloth over the water pots. Iron set to warm and iron over the cloth. A dab of white toothpaste on a damp cloth and gently rub in a circular motion.