Long story short is, the only way you’ll get the table uniform is to refinish the entire top. Doing that can be a dog and pony show, frought with potential for problems if the veneer is thin, sanding through being the largest of the problems.
So essentially you don’t have a lot to lose by trying a few things.
Is it Ok to put something wet on this top? Or does the simple presence of water cause issues? Not a flood, just damp wet.
If water is ok, damp a rag, and clean off the top where it is discolored. Then lay a towel, shop rag, whatever. White and clean are the criteria. Put the towel over the damp spot. Take an iron, with a steam setting. Place it over the towel, which is over the stain. Move the iron over the towel, and gently heat it. Take a look? Is it better?
If not, place a bit more water on the top to re-moisten it, place the towel back over it, and re-apply the iron, this time giving it a few shots of steam as you move it around.
Better now? If yes, quit, let it sit a few days. Still better, if yes good to go. If not quite right it can be re-done.
If it is good to go, the finish is probably a Shellac, great finish in general, but not quite “sturdy” enough for a kitchen table. You may want to refinish the top with something harder like Poly, it would give it a lot more protection. But for placing HOT items on wood, it’s best to use a trivet type deal, something with feet so the HOT is away from the wood a bit. Using a hot pad, a lot of heat carries right through it, and a Shellacked finish will discolor either whitish, or darker like yours is.
The other fixes for stains on most anything is rub either peanut butter, toothpaste, or Mayonnaise on them. I still have no clue what happens, but it often works, and if it were my table I’d go there first. None of them are likely to add any harm.
-- Think safe, be safe
