Components like hard drive(s), circuit boards and
their components that go into hardware device(s) have
MTBF ratings of approximately 100,000 hours.
The rated component life (hours) for the drive in the Apple TV is 20,000 powered on hours. (Refer to
Fujitsu | MHW2040AT | Full Specifications.) Many electronic components age more quickly at elevated temperatures, particularly certain types of capacitors & mechanical assemblies like platter motors & stepper armatures. As you approach maximum rated temperatures, you must de-rate components, both for power handling & for time to failure.
Believe it or not by powering a device on/off or
putting in standby, you actually lower the MTBF.
It is not that simple. One contributor to failure is high inrush (start up) currents, characteristic of certain devices with non-linear impedance curves. Induction motors & tungsten light bulb filaments are examples. However, this can be mitigated by "soft start" current-limiting circuitry. (This is commonly accomplished in drive motors.)
Another failure source is mechanical stress caused by thermal expansion/contraction cycles. While you might think that this can be minimized by avoiding power cycling, in reality electronic components experience large thermal swings due to normal duty cycles, for example when a CPU decompresses a chunk of H.264 content, then idles until another CPU-intensive task is scheduled. Think of it like stop & go traffic vs. constant speed driving: the former is much more stressful for engines & associated components. It's the same for computers, but on a much faster millisecond time scale.
Basically, the best way to avoid failure is to balance power on & off
intervals -- IOW, don't power up & power down so frequently that startup stresses outweigh the benefits of stress free, no power periods.