I would apply another colored transparent finish on top of the existing finish. I believe I know the type of chest you have.
Are the water spots white? If so, brushing on a little laquer thinner may remove the spots. Don't brush it just sort of float on the laquer thinner and see if it will remove the white.
Then brush or wipe on either an antiqueing glaze or a coat of varnish colored with a blue pigment ( blue will combine with red to make black...so be cautious) and let it dry and see what you think. Wipe it off of the copper before it dries.
That is known as a quick and dirty fix.
Linda C

Linda, quick and dirty is exactly what I had in mind!

The spots aren't white. The wood actually darkened a bit where the spots are. I'm mystified as to how they got there because they appeared recently. The only thing I can think of is that someone left the window near the chest open, and some rain blew in. Anyway, I don't mind the "imperfections" enough to go to great lengths to get rid them. The real issue is the redness of the finish.

The technique you describe sounds pretty easy, but I have two questions:
1) In using the lacquer thinner, should I wipe it off right after floating it on?
2) I'd love to have a black-ish finish on the chest. When you say to be cautious, are you saying that I should go light with the first application, then re-apply if I want more depth to the finish?

Thank you!

If the spots are dark, they are probably stains in the wood. White stains are water marks in the finish. You are unlikely to get a quick-and-dirty approach to removing the dark stain without stripping, bleaching, and/or sanding.

If you want to neutralize a red, you need the color opposite, green. Raw Umber is a typical color that you would want to use. Unfortunately, if you go to a big box store, you are unlikely to find anything called raw umber, but more likely something like Tavern Pecan, Rustic Maple, or anything else the mfr wants to call it.

What I would experiment with is trying to determine what, if anything is on there now.
Applying a raw umber glaze. A real paint store can mix you a raw umber universal tint in a glaze base. The result will be a brown. If you want a blacker finish, VanDyke will result in a darker color.
Top coating with a compatible finish to current finish. Shellac or varnish would be a good place to start.

I would try these in an obscure place (e.g., back or bottom) before starting on the top.

To answer question #2, yes, you can always go darker, but it's not always possible to go lighter.

Denatured alcohol would be a much safer starting point to remove white water spots than lacquer thinner. LIGHTLY DAMPEN a clean rag and with sweeping motions lightly pad the surface.

My technique for removing water spots with laquer finish is to float some on and let it dry, don't wipe or brush....but it won't work if the spots have gone through the finish.
What K says is good....
As to the blue or green to neutraloze the red color....depends on if the color is red to the orange or true red. Green deadens true red, blue kills orang-ish colors. If you want to knock the orang out of a stain, use blue...if you want to kill the red use green.
I think it is best to go darker in steps....lighter and then another coat. Not easy to remove too dark.
Linda C

Hm. I may be going in over my head even with this simple project. I will digest this advice and snoop around my favorite high powered paint store.

How can I determine what the current finish is? The wood has the appearance of almost no finish at all. There is no glossy look or slick feel that I associate with some finishes. It's almost as if the wood was sanded with very fine paper, oiled, then buffed.

the link below gives you some tests to try to determine the finish

Here is a link that might be useful: what finish is it?