It depends on what the finish on the mahogany is.

If it is shellac it will be very hard to remove paint without doing damage.

Lacquer is a little more resistant to damage, and varnish the most resistant.

Try rubbing with a rag dampened with paint thinner.

If that does not work, the next step is finding out what kind of finish is on the furniture.

Shellac can be disolved by using alcohol(denatured). Try rubbing arounjd a less obvious part with a rag dampened with it. Look at the rag if it is not obviously affecting the finish. You may see dirt and no finish.

Lacquer thinner is next(more expensive so last). Same technique.

Both shellac and lacquer can be added to existing like finishes and blended.

If it is a streak of paint on the surface of the finish, it can be buffed off with some 0000 steel wool and wax. If the finish has been scraped off as well, then it will also have to be touched up. There are tradesmen who specialize in this type of work. You can purchase stain pens and small quantities of brushing lacquer, but as an amateur you run the risk of making the situation worse.
Maybe just getting the paint streak off and wiping on a bit of dark stain would ameliorate the problem?
Casey

I agree with Casey. I do touch up work and see this a lot in moving claims and at that 2" line from the floor where the vacuum cleaner hits the baseboard, then hits the furniture. Most of the time, a light buff with 0000 steel wool does the job. Just watch so that you don't start removing color from the wood. Stop and get some help if you see this.

If the sheen does not match when you are done or there is obvious finish loss, pad on some shellac or spray on some aerosol lacquer.

Thanks so much for your input! I really appreciate it.

Sharon