What would happen if you used a copper penny to replace a melted fuse?

Well, first off most cents are zinc now, but in any case using a coin to replace a fuse is a very bad idea. A fuse is designed to be the weak link in an electrical system. It's intended to fail before any other part of the system does, so that expensive things like stoves, TVs, and inside-the-wall wiring don't fail in the event of a serious electrical problem. And even more important, a failure of one of those items could cause a fire!

Bottom line, keep some spare fuses on hand, or if possible have a licensed electrician replace the fuses with circuit breakers. They're special switches that open up during a over current fault or overload so they protect the system just like a fuse, but they can simply be reset after the cause of the overload has been fixed.