I'm having this same problem. I can get games to work occasionally, but it's very rare. At the moment my Wii is virginized, no loaders or softmods on it.
However, I just get the 'can't read disc' error, it doesn't tell me to turn off the system (unless I keep trying to get a disc to read - I've noticed this with my retail copy of Metroid Prime Trilogy).
I should mention that I've never been able to always properly read backup discs, I've always had to keep inserting the disc until it randomly worked. I've been using Verbatim DVD-Rs. Also, when there's a disc in the Wii (when it reads it) it's very noisy. Not sure if that's typical of the system.
My Wii's a launch system I think - LU50, I got it in February 2007. I've tried the cleaning kit a few times, and it took a small amount of dirt out, but didn't solve the problem. I'm not sure if I should send the Wii in for repair or not.