But my wife is stingy with the thermostat and my two PCs keep my computer room nice and toasty! 
Besides, I assuage my conscience my running protein folding so that Stanford can find a cure for Alzheimers Disease. (Disclaimer: Your computer will use much more power if you’re running folding@home than it will when idle.)
If you leave your computer on 24/7, anything that moves (like the CPU fan and other fans) will wear out much faster. A dead CPU fan can be fatal for most modern computers. You are also leaving yourself open to damage from things like brownouts and power spikes. Since most computers tend to run a little hot, running 24/7 will tend to shorten its life, especially if it’s in a warm environment like a room without air conditioning in the summer.
If you turn your computer on and off, then a different set of things is trying to kill it. The most prominant one is called “thermal cycling” which is basically where different materials tend to seperate when heated and cooled. This causes things like solder joints to disconnect (especially if it wasn’t soldered correctly the first time) or the itty bitty wires inside the chips that connect the silicon to the outside world will sometimes tend to lift up off of their pads and cause the chip to fail. There is also a thing called “inrush current” which is basically the extra current that goes into something to start it up. Since many devices require more current to start than they do once they are completely powered up this can cause a bit of extra stress on certain components.
Which set of effects will be worse for you depends a lot on how your specific system was designed and the environment that it is in.