How can I remove white marks from an old mahogany sideboard.

Try the majic eraser with some sort of gentle cleanser.

Are the marks scratches or just where the stain has faded?

Is it mildew or cloudy polyacrylic finish, or cloudy wax. If its cloudy finish, try moving a hot hairdryer over the surface back and forth. If that doesn't work, lightly sand with 400 grit sandpaper, then use 0000 steel wool and paste wax.

If its mildew, wipe down with vinegar on a rag, dry with soft cloth, and maybe put in the sun outside for an afternoon to dry it out removing drawers/opening doors so air can flow through.

If its wax, try buffing it out with a cloth or some 00 steel wool to remove old wax, then use 000 steel wool and paste wax to work into the wood and buff.

Try mayonnaise mixed with cigar or cigarette ashes...

Swirl Away works great! I searched all over trying to remove water marks on a cherry wood dresser. After trying many things with no luck I had someone tell about trying Swirl Away. You can find it in an automotive store. Its used to take out marks left in a cars finish. I put a bit on and let it set for a bit then used a clean cloth to buff it. It took a few times of replying it but in under a half hour the marks where gone. Try it first on a small area in case it might damage the wood, even if its a spot that has no white marks just to see how it does. I have used this on an old buffet, a mahogany end table and a cherry wood dresser with great results. I did find it at this link also.... https://www.walmart.com/ip/Meguiar-s-Swirl-Remover-2-0-16-fl-oz/23344083?wmlspartner=wlpa&selectedSellerId=0&adid=22222222227017323921&wl0=&wl1=g&wl2=c&wl3=40839192392&wl4=pla-78652269392&wl5=9017054&wl6=&wl7=&wl8=&wl9=pla&wl10=8175035&wl11=online&wl12=23344083&wl13=&veh=sem

White (not gel) toothpaste and a damp cloth. A dab of toothpaste on the cloth and gently rub in a circular motion.

Hi Janie, Restore-A-Finish works really well. Just wipe on with a soft cloth in direction of wood grain and wipe off with a 2nd dry cloth. It comes in colors to match most woods and costs around $9.00.