I'm not too worried about heat. We're putting it in our office, and I have my computer on all day which gives off some heat. A window that isn't insulated very well (Bah, renting -_-). And since we live in the California valley, it gets pretty hot. That on top of the UTH. I guess covering up the back wouldn't help. It's not like it's going to save a whole tree if I don't use the extra paper.
And for the tub, I wouldn't think you'd need to cover it up. I've seen many not covered.