You might just test with some DNA in a spot that is not visible and see if it looks like the finish comes off. If it does, it is probably shellac. On one table I refurbished, I simply used some DNA on a cloth as a cleaner which takes off a little of the top layer of shellac along with any grime that has become sort of embedded. Then, do some light sanding to smooth the surface and apply one or more coats of shellac. You can top it with paste wax.
If it looks like it is varnished, I would probably try something like Murphy's Oil to clean the surface and then simply apply a paste wax.
If it not shellac, lacquer or varnish, you could simply try applying tung oil, pure BLO (no varnish) or maybe Howards Feed and Wax, just to freshen the finish. Experiment on the back side of a leg or an inconspicuous spot to see how it looks.
--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.