Well I guess you could, but it will make the quilt thicker and heavier, and harder to quilt through. Usually once you sew the applique on, you cut away the background.
Regarding the class that recommended putting something under the applique pieces - I've done that when my applique is a lighter color than the background and I'm not planning on cutting away the dark background. It's done so you don't see the dark background showing through the lighter applique.