Does anyone know a way to get tung oil out of antique oak wood?

An oil based polyurethane may not cause problems. Tung oil usually provides a nice finish and can easily be brought to a new finish level with application of new coats. Make repairs as needed to the wood either filling or raising and sanding or scraping, and then apply the oil. Likewise Poly can be applied in as many coats as you care to apply to get the finish you want. Sand between coats and wipe with a tack cloth. After a minimum of 3 coats rub with 'rottenstone' which is a very fine abrasive you can get at better hardware or paint stores. You mix it with a little water and gently rub to a highly smooth surface to apply your final coat of poly.

Poly will never look as good as an oil / wax surface because it 'sits on top' of the wood, not soaking in, but the poly is very durable.

Please practice on scrap or the underside to ensure you like the results.

Your original question on removal - you can put the table in sunlight to warm, perhaps creating a plastic greenhouse around it for higher temperature. You would then wipe off the table and repeat until the wood stays 'dry'. A long process unless you are in the southwest!

You can also use Oven Cleaner or Engine Cleaner to wash and dissolve oil again until the wood is dry of oil, then you will need to finish the wood with sanding and scraping etc. before you stain it if desired and protect it with the poly. I do think oil based poly produces the better finish.

Please, please don't destroy the top of this table.

If the tung oil is dry, then you can put polyurethane over the top. There should be no problem, I do it all the time.

Okay, I accept that from one Cunningham to another.