Gah. It looks like it is the game that's duff, but in some strange way that means it works on the Wii at work but neither of the ones I've gotten. I guess it's better for the game to be bad than the Wii, but I really don't know what could be wrong with it. There are no obvious scratches on it, and the error isn't even a disc read error, just this painfully generic one. Very frustrating.
EDIT - Got a replacement Mario. Things are looking better now. Although frustratingly I tried to copy across the Miis I saved on the Wiimote and now it says I'm not allowed to edit them because I didn't create them. So you have to copy them slowly by eye, or just start over. It's like DRM for kids. Weird.