You should give this a try. Get some copper and get to it. I wanna see the end results. If it works good you can market it as a passive cooling system. Reserator style. Then all sorts of people will want to by it. (as indicated by a recent trend in several particular posts at this here site) You can even come up with a brand name. Perhaps Kronchlance.
But seriously, I have been interested in building a case out of copper for quite some time. I always think of it as the next step in the evolution of the case. Of course there are all kinds of funky design considerations that one would have to work out for something like this. Could you imagine how bad ass it would be to have a copper case with the tubing like you mentioned placed inside hollow copper walls. Unfortunately, copper is harder to come by then aluminum and steel. Maybe instead of using tubing, you could use a length of hollow copper bar stock. Of course that will be a hard thing to find.
ack, this has got the brain going on all sorts of passive cooling ideas that use the case as a radiator. I am going to have to steel your idea.