Depends on the type of system. For electronics that have many discrete electrolytic capacitor components in close proximity to heat-sinked components, like power supplies, after years of use the electrolyte could dry out in the caps, causing their ESR to increase with the result that, if powered off and left to cool down, the circuit would not fire back up. It was considered best practice to leave such devices powered on.
Mechanical devices like spinning HD platters will eventually fail just because the bearings have a finite life time.
As for flatbed scanners, the electro-optics are probably best left powered off when not in use, though the mechanics like the rail that the scanning heads rides up and down could in theory bind up through disuse if the lubricants dry out. This probably has more to do with your local climate. I'm in a dry, high desert climate where drying of components through disuse is a real problem.
Also, dust in your home environment can affect these mechanical systems. I've seen many problems with optical drives (CD, DVD) where the rod that the head rides along gets contaminated with dust, and the lubricants dry out, causing the head not to track properly because it sticks and binds on the rod. This is especially problematic with many of the designs that use a linear drive rod like what flatbed scanners use.
In the early days of CD technology, I recall that the Philips brand were more reliable than the Sony because the Philips used a pivoting mechanism to drive the head between center and edge of the disc, while the Sony used the linear drive rod method that more easily would bind up due to dry lubricant and dust build up. That was many years ago, so I don't know if that's still the case with those brands today. But keeping your scanner clean and dust-free, in a normally humidified environment, is probably more important than whether it's left powered on all the time.
-Joe
