Storing items in washroom (protection from moisture)
You can run a test with some scrap paper or something, but I suspect this won't be a problem. Towels, toilet paper, and tissues are all commonly stored in bathrooms with no ill effects, and many vanity cabinets are made of particleboard - notoriously susceptible to moisture. 

Just make sure to run the fan every time the bath or shower is used (don't wait for it to turn on automatically) and for 20-30 minutes after.