I've heard this all the time as well, some people say that when you turn off a PC all the fans turns off immediately, causing the components to overheat. I've heard that turning it on is the same as x amount of hours/minutes as if you left it on.
IMO, I would only leave a computer on if I knew I'd be back within a couple hours. Anything longer then that I usually just put my computer in standby, or turn it off completely. I've had computers that were 10-15 years old and still work, so I don't believe that it would actually wear out before the computer is useless anyways.
It really comes down to whether or not you want to pay for the electricity or not. But remember, leaving a computer on for 8 hours overnight wears it out as well.