Any treasured wood chess set is ideally kept dust free on a regular basis. At least once a year, it's a good idea to wax and buff using a paste wax sparingly. Allow it to dry before using a cotton cloth or similar to buff lightly.

Avoid liquid polishes, be sure to use wax sparingly, and remember to allow to dry before buffing. 

The wood chessboard can be polished with a good quality furniture wax. Avoid wiping the chessboard with a dry cloth unless it is extremely soft material such as camera lens wipe.

For all wood pieces and board, avoid liquid polishes and silicone base since this can damage the finish.

Generally follow the same care tips as you would for fine furniture finishes.