I'm pretty sure that is mostly rumor. I believe in previous version of the firmware there was an issue with the wii connect causing heating issues, but I never heard of anyone losing a wii to overheating. The hardware is made very well.
If you're still concerned I'm pretty sure you can turn off wii connect through system settings. I read someone saying that the new wii firmware (after doing system updates) solves this issue and it should be fine now.
1. I don't believe they are true. I've left my wii on with wii connect 24 for a year or so.