Walls – How to mount a white board on a wall without making holes in the wall

Mold needs three (or really four) things to grow: Stagnant air, moisture/humidity, and dim or dark areas with organic 'food'. The solution you're looking for probably involves using a permanent waterproof barrier (either a paint product -- which would be an industrial coating -- or a foundation product) in order to largely remove the humidity from the equation. Those coatings would need to be installed by a professional. As a cheaper and DIY solution, you might also consider 6mil plastic vapor barrier on the back side of the wallboard.

Run a strip of masking tape on the wall where the molding will cover it up, but where your nails will hit it.

I hammer nails into old plaster all the time by putting a piece of masking tape on the wall and hammering into it. It holds the plaster together while the nail hits it and can easily be ripped off (gently) after I'm done hitting it. In your application, you can just leave it hidden behind the boards!

Use masking tape because as, for example, duct or painters tape dry they grab onto the wall harder. I have pulled large chunks of plaster out by trying to remove old duct tape. I have successfully pulled fresh duct tape off of plaster walls, but I don't recommend trying.