You are correct, they are not allowed. Some ships will do the ironing for you, and I know that Princess ships have a laundry room that yuo can wash and iron clothes in.
You will not find irons or ironing boards in the cabins, as your assumption was correct: Irons are a safety (fire) haZard. (One Z...not two. Just be aware of that before you find another poster's spelling "ear-a-tate-ing"). Thanks.