I would paint it. It helps seal the wood from moisture and makes it easier to wash down the walls when you need to do that. Chickens won't mess with the paint. Just wait to put chickens in until after any paint smells are gone.
I don't know why it would be a problem. Using something that's as durable as possible and that holds up to washing the best would last the longest. Other than that, nothing's really a problem. You don't even have to paint, it just helps.