Will “Magic Eraser” take water marks off my wood furniture?

No. There is a method where you put mayonnaise (I’m not kidding!) on those spots and let it sit overnight, or something like that, and those marks will be gone. Wait for more responses here to get more ideas but a magic eraser will damage the finish for sure.
I recently worked a spot on my kitchen table with Goo Gone and it worked fairly well. A furniture polish like Old English 800 might work well…

It will tear up your finish.

I heard that letting rubbing alcohol soak in overnight will work as well as mayo. One of the many things I learned the last time I was visiting Kansas.

When the hell did you come to Kansas and not come see me???

Hand sanitizer / rubbing alcohol takes permanent marker off of stuff. Not sure what it might do to your finish though.

Last time I was in KS, it was before I discovered Fluther. I picked up a newsletter at a diner, and there were tips on touching up furniture. The most interesting one was the use of rubbing alcohol to remove water marks.

There are different strengths of rubbing alcohol. Some is 99% pure, while other brands are 70% pure. If there is any water in it at all, it’s possible it would make your watermark problem worse.