Step 25: Appendix: Non-toxic waterproof wood finishing
If you go with tung oil, note that there are lots of "tung oil finishes" that contain little or no tung oil, and use the name simply to mean it is a wipe-on finish, which may include many toxic ingredients. So be sure you are buying the real thing.
For an added layer of protection, you can add a few coats of shellac, which is a bug secretion applied like a paint by dissolving it in alcohol. Normal shellac doesn't do well in water--it turns white. But de-waxed shellac does much better in water. You can buy flakes of de-waxed shellac and dissolve them in denatured alcohol that you can buy at your local hardware store. Shellac.net sells a wide variety of shellac flakes and has further information about shellac. Unlike many other finishes, shellac sticks well on top of tung oil, at least if the tung oil is fully cured first (which might take a week or two).
I doubt the tung-oil/shellac combination is as waterproof as a good marine varnish, but it's kept my shower pedal looking good through daily showers for at least six months now.