You need to sand them to bare wood, re-stain, and then apply a film forming coating like polyurethane. And next time put down dropcloths and wipe up any paint drips immediately.

There is absolutely no need to sand to bare wood.
Get all the residue off with mineral spirits then a light sanding with 220 grit just to rough up the surface. Then apply several coats of poly.

Fabulous, thanks for the advice Don.

Depending on what 'goof off' prduct you used, a number of them use Xylene as the main solvent.

It will (at least) damage the gloss of many finishes, and damage other badly.

If the damage is confined to the clear finish coat, screening and re-coating may be successful.

It is likely worth trying over fully stripping.

You should be able to tell after the first new coat.

the damage does seem to be confined to the finish. Actually the spots that aren't glossy are very hard to see, and have to be looked at in a certain light to recognize them. Out of an abundance of caution, because this is a bathroom I want to make sure that I have a good seal on there. The wood itself looks fine.

h=How will I be able to tell after the first coat? What would be my indicator that it is or is not working? Sorry if this is an unitelligent question. I haven't worked with hardwood floors before. Thanks in advance!

"How will I be able to tell after the first coat?"

Allow the first coat to dry its re-coat time and see if you can tell if the spots are still visible.

They should be pretty much gone after screening since it roughs up the while surface anyway.

If it was old fashioned varnish you could probably just rub the dull spot out, but poly finishes often do not respond to 'rubbing out' well.

Poly does not touch up at all well. The entire floor will need to be sanded and recoated, and you will still see the line where you stopped at the threshold. In addition, some of these paint remover products or floor cleaning products that you have used will impair adhesion of the new to the old. If that happens, it may look OK at first, but then start peeling off in a couple of months. If that happens, the only recourse will be to sand to bare wood and start over completely. If you've ever used anything with silicone or orange oil in in on the floors, skip the intermediate peeling step and sand down to bare wood now.

the Goof off I used contains Xylene. So, I'll try sanding off the finish in the entire room and recoating with poly. It's a pretty small room so this won't be a big chore. Thanks everyone.

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"Poly does not touch up at all well. The entire floor will need to be sanded and recoated,"

Screening may be adequate.

Have a local floor finisher take a look.

It is not "touching up" to screen and put down a while new coat.

You can use the Red Scotch-Brite pads. Go over the complete floor to scuff it up, then recoat with a good quality of polyurethane. I like the water based poly for the fact that odors are not as bad, and it dries hard.

Terrific, thanks Kevin for the specific instructions. Much appreciated!