The same thing happened to my first Wii, and I sent it back to Nintendo. A couple of weeks later, I got it back, but it was actually a new one. They included a little checkbox-form thing where they had checked the box saying that they couldn't replicate the problem, so they just gave me a new one.
If this happens:
They don't transfer your console's Wii code (you will have to distribute the new one to everyone again).
I THINK friend codes for individual games stay the same, but I'm not sure because I don't think I had any online games at that time.
They do transfer your saves and channels. (I wouldn't worry about the ones that you can't copy to an SD card; I'm pretty sure they just copy the entire contents of the flash (or whatever it is) directly.)
You have to re-download your VC games. (Presumably because of the console-specific encryption.)
Another thing you should keep in mind that I didn't know about is Miis. They get transferred over to the new system, but it won't recognize that you created them, which means that you won't be able to edit them or send them to anyone. So if you care about your Miis you'll probably want to store them on your Wiimote or something.
By the way, after I got my new Wii, a little while later I started having the same problem again. But then I stopped playing for a few weeks, and the next time I tried, it was back to normal. I don't know why, and I haven't been able to get an explanation from Nintendo about what actually causes this (so I could try to avoid having it happen again).