When WiiConnect24 Is on, it heats up because the fans don't run, but it is still using its network adapter, and parts of the mainboard. If WiiConnect24 is off, it stays cold.
From what I understand, the fans don't turn on. And yes, mine got very hot too with connect24 turned on. If you're concerned about heat issues just turn off connect24 and place your Wii vertical. Problem solved.
how many times are we going to see this thread. obviously all the heat is generated on the left side of the console when it is in its little stand. so when you flip to to sit horiztonally, the hot side is now on the bottom, thus no way for the heat to escape without a fan.