I have yet to see a computer die because it was turned on too often. The processor chip and the RAM don't need to be lubricated in order to function. Two parts that do wear out on a computer are internal hard drives and mice.
Hard drives fail for a variety of reasons - from broken belts to getting scratched by dust.
Startup is just like any job for these and other parts, and being shut down is a relief if nothing else. What really stresses a computer out is dirt. Dust and cigarette smoke account for most of the hardware problems computers are prey to. It collects under your keyboard, on the rollers in your mouse and on the magnetic heads in your disk drives - hard or floppy. Of course, lots of computers in the world never are turned off.