What should I do about the tired finish on an antique, Mission dining room set?

furniture that first appeared more than a hundred years ago. Its features include square design elements and slats. Quartersawn white oak was usually used to make Mission furniture.

The only proper way to assess the value of the current finish is to have it examined by a skilled appraiser. I wouldn’t attempt any kind of refinishing until you do that. What portion of the finish is now bare wood? If it’s more than a little bit, the existing finish probably isn’t worth too much. A skilled furniture restorer can repair an old finish without stripping, though you’d want to ask for references and check them diligently before trusting your dining set to anyone.