Damm, you have nothing to worry about. The Gamecube is a TOUGH machine, mine stands as evidence to that. I put my Gamecube through years of abuse and impractical use. My Gamecube has been left on, literally, for weeks at a time. (It was for some game, I can't even remember which one it was now.) And the most that happened was that it got slightly warmer than usual in my game cabinet. (Note, I had the Gamecube CLOSED up in said cabinet for these weeks. Something I fear to do with my 360 for more than a few hours.)
So, unless you're going to leave your Gamecube on for a span of several months, I would not worry one bit about overheating. I have never in my life heard of an overheated Gamecube, and I think that it is safe to say after what I put my Gamecube through, the chances of a Gamecube overheating is slim to none. As for a chipped system, the soldering used for console modifications is not far from the industrial-grade stuff that Nintendo uses. At the hottest my Gamecube has gotten, there's no way it would mess up a modchip's inner-workings.
TLDR; Gamecubes don't overheat, modified or not.
