how can i clean an old Georgian mahogany dinning chair that has become very dull and greasy with use?

Answer 1

Use a cloth soaked with white spirit to get the grease off. Polish off with a dry cloth, and then, using another dry soft cloth (or soft brush), put on some dark brown boot polish. Evenly spread on and polish off to leave a brilliant shine and an minor scratches filled in. If there are any bad scratches, you must fill using a proprietory wax filler and repeat above minus the white spirit. I do this for a living.

Answer 2

Murphy's oil soap and a soft cloth. And don't wet it too much. Make sure to wring it out well - no drips, either.
Put a good furniture polish back on after it dries.