I covered my sleeve boards with muslin and the ironing board with unbleached cotton duck. All-cotton twill is also good, and there's a fuzzy-back twill called "canton flannel" that makes a good ironing surface -- particularly when you throw a cloth over a blanket on a kitchen table to iron yard goods; the fuzz keeps it from sliding around.
Avoid dyed fabric, printed fabric, and synthetic fibers. (Nomex is acceptable, but costs more than cottons that are at least as good.) (Presumably linen and hemp would be better than cotton of the same quality, but these fabrics are not easy to come by, and decent quality is very expensive.)
For padding, wool batting (or an old blanket) is best, cotton batting is good, polyester batting is Right Out. The old padding is usually re-usable, but should be tossed if it's lumpy.
I recently learned that cotton cord is better than nylon cord for drawing up the edges -- the fullness stays where you put it, and doesn't loosen up again while you are tying the knot. It does require you to push the gathers to where they are wanted. I push too many gathers into the corners, then stretch some of them out while putting the cover on. This leaves very little gathering to do before knotting the cord.
Joy Beeson