The problem with varnish today is that the term Varnish is used to describe all sorts of finishes: polyurethane, urethane, low VOC, water based etc no telling whats in them.
Traditional varnish was a mixture of Resin (generally Copal), Oil (Tung or linseed), Metallic Driers, and a Solvent (Mineral Spirits, Naphtha, Paint Thinner, Stoddard Solvent).
So if you are using a traditional varnish; which is getting harder and harder to find, it should be fully compatible with tung oil, for thinner I would use Mineral spirits.
Try mixing some, and apply it to a piece of scrap wood first.
