I don't think the steam will affect the wood because you would have batting, a first covering (possibly muslin) and then your top covering (possibly ironing board cloth that has an aluminum finish or a heavy, close woven cotton like duck cloth). Your iron would have to be a heavy duty steamer to go through all of that.
You can buy ironing boards that have a fan mechanism in the board that will pull the steam through into the fabric of what you are ironing and then into the board. We used one in our costume department and it was terrific to 'set' seams , pleats and other required forms. They are really expensive but I would love to have one for my own personal use - they are great to use and very useful!