So would that mean the logic board gets continuous use even while asleep? Wouldn't that make its life shorter?
While the "logic board" is physically one part, there are many many distinct components on it. Your graphics card, for example, would have no reason to receive any power whatsoever. Your CPU would have no reason to receive any power.
Regardless, we're talking electronics here--not mechanical parts. Using a car causes it to die. Using a hard drive causes it to die. Running small amounts of electricity through metal does not cause the metal not to be metal anymore so long as it's not enough electricity to melt the metal (see incandescent light bulbs and fuses). That's the key detail here. We have satellites at the far reaches of the solar system that have been running for decades with absolutely no way for anyone to swap parts. On the other hand, there are no power surges from lightning strikes, and there's no ESD from people touching them. They do have to deal with radiation and micrometeorites, but after several decades of electricity running through the components, they're still running.
Even when your computer is "off," and unplugged, there's still a clock running. If it's "off" and plugged in, it's still capable of turning "on" based on a schedule, which means there's circuitry running that's thinking about that information. It's still running electricity through the power button. It's not an easy task to cease all electrical flow. What you think of as "off" isn't really.
What will make your machine last longer is a good surge supressor that tells you when it's not doing its job anymore, a UPS for brownouts and blackouts, you being conscious enough to physically unplug everything when there's an electrical storm on the way, and an environment that's cool enough to help the machine prevent itself from overheating.