I always wash quilts when they're finished, too. It softens and fluffs them and removes all the dust, skin oils and whatever might have gathered on it while you were working on it.
If you can take them to a laundromat where they have those giant front-loaders, you'll probably be happier with the results than if you use your home machine. They seem to get the quilt cleaner - be sure to use a few color-catchers, either way.
I would applique a small circle of the background fabric over the burn. It won't show once you get the quilting done. I had to do this in four places on the back of one of my quilts, and they're hard to find.
