Before I had a chance to freak out too much, my husband did some googling and we discovered that on some pieces of furniture if moisture (usually from the steam condensation on a hot dish) gets under the finish, white splotches called heat marks will appear and will stick around until treated.
You'll be surprised to learn, as I was, that to remedy heat marks, you apply... more heat and moisture! Weird, right? I just used two folded over clean white linen napkins and sprayed them down a little with water. You want there to be several layers of cloth between your iron and the table surface. Then I ironed over the napkins on top of the heat marks for about 15 seconds with the iron on a medium heat/steam setting. Don't just set the iron down though - keep it moving.
I have no clue why my table looks so dark here. Must have messed up the camera settings.
It was like magic and the heat marks disappeared (though in these photos I can see I missed a spot that doesn't show in person).
**A little disclaimer: this trick worked for me, but I guess a few other readers have not had success. It might be wise for you to try a less conspicuous area on your table first. Be sure you're using damp cloths with your iron on a medium setting with steam. And be sure to keep your iron moving!**