No, it will just say it doesn't recognize the disk and if it DOES recognize it it would just reset when you tried to start the game.
UNLESS
You install DarkCorp. Then you can run them from the disk channel, too.
DarkCorp replaces a lot of your IOS's with patched versions so many say it's a bad idea but as long as you don't power off your Wii in the middle of installation you should be fine. Even if it does there's a good chance it'll be fixable. (no promises, though, like anything with Wii hacking).
It would also be recommended to have PriiLoader installed and have it block updates if you don't already have it. That's the only real way running a disk from the disk channel can cause a brick. The region free PriiLoader hack is nice, too.
EDIT : Sorry, that's for Wii games. I didn't see you were talking about GameCube games. What would happen is the Wii would just reset. No other harm done. the original cMIOS needed to have a GC backup loader run first to be able to run games. There are versions of the cMIOS that have the backup loader built in, though, that you CAN use to to load from the disk channel. You can download a WAD to install three different versions of the cMIOS with ModMii for Windows (download page 4 middle of the right column). 2 of the 3 options work from the disk channel. I would recommend the bottom one on the list (the one marked as #5) since it's more stable.