I just shipped my Wii to them a few weeks ago. They received it a few days ago and I guess they're repairing it now. So I guess I have half-experience.
From all that I've read, if it is a problem that is fixable by replacing a part, tightening something, etc., then they'll fix it. If it isn't like that, they will transfer your data to a new Wii and give you a replacement. Either way, they will do whatever they can to save your data. It would make sense to me if they backed up your data before messing with anything internal, but I'm not sure about that part. I backed up all data that I could before sending my Wii in (game save data mostly). You don't have to worry about VC games because you can just download them again. Well...you shouldn't have to worry about anything at all. If Nintendo didn't care about your data, they wouldn't have such a highly rated customer service.
I hope that helped!