I'd be more concerned about my quilt than the washer. Papers are treated with a number of chemicals in the production process - then add the ink - BLECK! I'm not willing to risk my quilt for that.
I have used light weight muslin, it worked beautifully. In fact, one of my favorite throws is a string quilt on muslin with no batting and a heavy flannel backing. It's the perfect weight for me right now - cozy, but not too hot. Things get too danged hot all of a sudden every now and again...