Longer than most of us will live, if the part isn't defective. You also can't predict whether an electrical component will last longer under continuous use or with frequent heating and cooling cycles. So good luck knowing how to not "stress" your computer.
BTW, I've never seen a MTBF rating published for a microprocessor. I'd be very interested to see these numbers, assuming they exist. I would also doubt very much if a PC board has any MTBF numbers.
Actually, every little component will have a MTBF rating. Even the resistors, capacitors, and such.
As for leaving it on, or turning it off, both will cause you to reach the end of a given components life. It's just a matter of time.
Essentially, letting it cool, and then suddenly shocking it to life causes stress, and will kill it in time. Think about how your light bulbs tend to blow when turned on instead of when left on.
But, at the same time, you will reach the actual rated time faster if you just leave it on. So, that could lead to faster failure as well.
But, the MTBF rating for most computer components exceeds the 3 to 5 years that most computers are in actual use.
But, if you are one of those who will keep a computer for say 10 or 15 years, then you might meet those ratings in time.
I've seen office computers that were left on for 10 years. And, then one day they get turned off, and back on. And, they die on being turned on. Essentially, they were already weakened, and the stress of being left off for a while and then shocked back to "life" killed it.
Trying to predict which method of use will kill the machine faster, is pretty much futile.
But, you can be assured that leaving a computer on will cause you to reach the life expectancy faster. But, turning it on and off will cause more stress to the components than if you left it on. So, pick how you want it to die, and go for it.
I don't really tend to worry about it much.
But, since I used to design and build computer accessories out of raw materials (as in building it from components as my own PCB boards, custom cases, and hand soldering all the resistors, capacitors, chips, power system, etc.), I am quite aware that every component has a MTBF rating.
And, yes, PC Boards do have MTBF ratings. One of Intel's boards for the Core2Duo / Core2Quad chips has a MTBF rating of 218,279 hours (or almost 25 years).
http://download.intel.com/products/motherboard/DG33TL/tps.pdf
Intel doesn't like to publish it's MTBF ratings for it's CPU's. But, you can call them for the info if you want:
http://www.intel.com/support/processors/sb/CS-001666.htm
But, MTBF ratings are rarely what you can realistically expect. For example, most hard drives have carried MTBF ratings of 80 to 140 years for the last decade. But, hard drive failures are quite common, and obviously, they have not been in use for 100+ years.
In the end, the life of a component will be determined by it's weakest part.
But, in the real world, computers rarely see more than 3 years to 5 years of solid use before being retired for the latest and greatest.
In the world of Apple's, many people only use them for 1 to 6 months before ditching them for the next latest thing.
So, it's unlikely that most of today's users will ever have a machine die because it's useful life expectancy was met. It is more likely that they'll see failures do to weaker than expected parts.
What it takes to kill said part can be quite varied. It could be turning the computer on. Or, it could be allowing the component to get too hot (as in working it hard, or improperly cooling it). Or, it could be weak enough to die with no real cause at all (other than being a dud).