It doesn't make too much of a difference in the lifespan of the computer whether you turn it on or off.
I've heard though that it's best to leave the computer on rather than shutting it off too much since shutting your computer off and then turning the computer back on too much can cause wear and tear on the cpu and other things such as hard drives etc.
Things also wear out sooner by being heated up and cooled down.
Leaving your computer on seems to be the best to increase the lifespan of your computer.
I leave my desktop on unless I'm gonna be gone for more than a day.
But I do restart it once in awhile.