I'm wondering whether I should apply a finish to the wood.
Great question! I think the best place to start is with what other people have done. I did a little searching and found a few examples of old wooden ironing boards on sites like eBay and Pinterest, and a new Amish-made one on Amazon. The new one says the legs are finished but the top is not, "for safe ironing." The older ones also look unfinished to me, but it's hard to be certain because the patina that comes with age and use might have started out as some kind of finish. All the boards I saw look to be fairly flat, though, even ones that are quite old.
Ironing is going to create steam, which could introduce moisture and warp the board. But ironing will also create heat, which could take off the finish and stain the cover...
I think you're right to be concerned about the heat. I know that there are some polyurethanes that can tolerate up to 300°F, but the temperature from a clothes iron starts at around 300°F for low settings (e.g. silk) and goes up to 450°F or so for the linen setting. Even though there will be some fabric and batting (and clothing!) between the iron and board, I don't think most finishes would handle that kind of heat well.
The worry about steam seems less likely to be a problem. The steam introduced by ironing is pretty short-lived -- it comes in bursts, and it's unlikely to send moisture much beyond the surface of the wood. And steam can actually dry wood out -- by definition, steam is water that's above the 212°F boiling point, so in some cases it can cause moisture in the wood to vaporize and escape. But again, the steam from ironing comes in fairly short bursts and is very localized, so it seems unlikely to cause any major change in the board. If you're really concerned, you could add a couple of crosswise pieces on the back of the board to hold it flat (just be sure to use screws instead of glue, and to elongate the holes a bit).
If it were my ironing board, I'd leave it unfinished.
