I've been making several chalkboard table runners and chalkboard placemats as Christmas presents. And, even I can make mistakes such as hitting the front of the chalkboard fabrics with the iron.
Thus, I have come up with a solution. While you are constructing your table runner or placemat, ALWAYS iron from the back. Then after you finish quilting your piece, pin (or in may case, tape) a piece of cotton fabric over the chalkboard fabric so that you do not inadvertently hit it with an iron as you are applying your binding.