How should I handle bare wooden posts in a wet area?
I think you should seal the wood for two main reasons:


the posts are in a bathroom, where they'll be subject to high humidity (possibly condensation) and temperature changes. That's a recipe for warping and splitting in any wood. Even with the thick posts used in a post-and-beam house, you'll still see some ill effects eventually.
Doug fir is rot-resistant, but not rot-proof. If you do get condensation in that bathroom, you'll shorten the life of those beams if you don't protect them.


A third (cosmetic) reason is that untreated wood is going to lose color over time, becoming gray.  I would recommend you think about treating all the exposed posts and beams, not just in that bathroom.