Cleaning and Stain Removal - Wood Dresser Help!

I used a clean towel that had some white paint stains on it to iron on.

After ironing I noticed some white spots on my wood nightstand. I'm guessing this came from the paint. I don't know what kind of finish the nightstand has.

Please tell me what I can do to fix this. I wiped it down with a rag and water right away but the whitish stains still stood.

There's a new invention called an Ironing Board, LOL.
Firstly, I wouldn't advise ironing on furniture, but since you did, I'm guessing the heat melted the paint and then bled thru to the furniture. Have you tried some furniture polish or lemon oil and some elbow grease? Might do the trick. If that doesn't work maybe try some scratch cover.

Marks could be heat marks. Try rubbing in direction of wood grain with some tooth paste. Some apply mayonaisse (not Miracle Whip) and let set. If you have cigarette ashes available, these can be worked in with the mayo and used as a gentle abrasive, rubbing back and forth in direction of wood grain. You may also try a little ammonia on a white cloth and rub. If finish is lacquer, you can try rubbing with #0000 steel woold and mineral spirits (test first in inconspicuous area for ill effects on finish). If DIY attempts are not successful, then refinishing can be done.

it sounds like to me this has more to do with the heat than the white paint. The white halos are were the finish of the wood has been damaged. Twelevepole has it all covered.

If it's not the paint, then it's more to do with the steam (moisture) from the iron, rather than the heat. I use an iron almost everyday and any damage done from a hot iron results in scorch marks. I have also heard about cigarette ashes being good for water rings.