There's no definite answer to this that I'm aware of. It's a long running debate in computers/electronic devices.

Some say the constant turning on/off adds unnecessary wear and will kill the device sooner.

Some say the constant running/flow of electricity through all the components will kill the device sooner.

Do whatever you want.

Just leave it on.

I just leave mine on.

Actually, the main argument I've heard against turning off your computer in production is about moving parts like harddisks and fans. But most computers will still turn those on and off all the time, unless you configure them differently.

My current laptop has "conservation mode" for the battery, so that it doesn't constantly try to top-up the battery, which can also be tough on the hardware.

As for the solid state stuff like CPU and RAM, there's some build-up inside the circuits, but it's hardly ever a problem unless you overclock them, if you're concerned, you might be able to under-clock your device. If I remember correctly, the default power-profile on the Shield is "high performance", there's a couple of other profiles available I think, choosing a low power mode will also make the CPU less likely to run at full speed, that could mean a longer life for your device.

Along the same lines. The TV that I use for digital signage is plugged into a smart outlet. I have it set on a schedule to turn off at night and turn back on in the morning. Is it bad that the TV's power is just cut off and then back on the next day? Is this a lot worse than using the remote control?

I have heard and with the nvidia defect gpu problems experienced it myself, that when computer chips heat up during the use and then cool when turned off it can create movement due to the heat expansion and shrinking effect.

With that constant heating and cooling leaving your computer on could in fact be better for the longevity of the machine.

However things like that nvidia defect fiasco are rare and with modern computers that is not really a problem.

So with modern Android tv devices, I don't think that this is a problem. Only thing that is going to suffer is your wallet, due to the bigger electric bill.