Keeping WiiConnect24 on does cause the SD card to feel quite warm, even if the Wii is off.
I dont know and im not sure about this but when you close your wii, do you leave it at a yellow light or a red light? I think the yellow light is whats causing the problem since it keeps the wii partly on while its off, causing it to heat. Once i started leaving my wii off at the red light, everything went out alright (even when Wiiconnect is on).
Without WiiConnect24, you can't receive Wii Messages and other stuff. Many games use it, too, like Animal Crossing. It doesn't make the Wii heat up that much AT ALL... There must be something wrong with your Wii if it's getting overheated from WiiConnect24. And I rarely use my Wii to play GameCube games, so I don't usually don't have a memory card in my Wii, anyway. And when I did leave my GCN memory card in my Wii for long periods of time, nothing ever happened to it.
Your wii does overheat. It used to happen to me all the time until i left it at the right light, when i turn my wii off. My wii doesnt heat up any more.