Funny you mention this as I'm ironing some binding right now. I do the French bias method of making binding, so I fold the raw edges together as I press. I pin the heck out of it as I put it on -- raw edges of binding and quilt together, I sew on the top of the quilt and fold over and hand blind sew on the back. When I'm doing the hand sewing I don't use any pins or hair clips or anything, too many gouges in my thighs from the pins and just didn't like working with the clips.
I've never seen nor heard of anyone ironing while it is on the quilt, seems like a really good way to gunk up an iron if you use poly batts, but no issues I guess with warm & natural.