I cover mine in muslin ... Haven't tried washing the original. It's a board I bought way back when for pin weaving ... remember that ladies? It has wonderful gridded fabric but the back is a single layer of fabric and feels like there's some kind of board in there so I don't want to risk warping anything.
I cover my ironing board in muslin too. I'm lazy so I just safety pin it to the original cover and when it gets dirty I unpin and pin on a clean one while the dirty one goes in the wash.