Although modern electronic equipment seldom contains thermionic valves (or tubes across the Atlantic) which were prone particularly to their (cathode) heaters burning out similar to typical tungsten lamps, the majority has Switching Power Supply units built in, or in the lighter-weight external, non-transformer, mains adaptors.
When the device is switched on manually by the user, the working DC Voltage level is achieved by switching the 240 V Mains supply on and off rapidly (many times per cycle), almost continuously for varying periods, so that the DC Voltage on the load supply rises to then hovers around the required working level.
Also the User Swich-On can be at the rhe peak of the mains wave-form at 1.414 times the nominal 240 V RMS (= about 340 V; and the mains supply in our area has been as high as 260 V RMS so almost 370 V instantaneous), or could be at about 0 V (ideally) as the mains waveform transits between the positive and negative peaks of the sinusoidal waveform. Or at any voltage in between, following the sinusoidal waveform.
So there can be quite a lot of electrical "noise" generated at varying values and frequencies.
This noise could simply "swamp" some of the settings; or even on very exceptional occasions, generate apparent instructions etc.
At an extreme, the noise could actually "burn out" or physically destroy the semi-conductor circuits.
So electrically/electronically it is generally better to leave the devices switched on.
However, it is occasionally necessary to swich them off for say about a minute, so that a clean start is made on the settings or even to update them.
Some of you may be aware of having to do this with Sky Boxes and the like; or defragging a hard disk etc. Defragging can physically harm the typical HDD, so should not be done very often; but is recommended on a periodic basis.
On the saving of electricity aspect, all of the power "used" by the devices ends up as heat; and as generally some degree of home-space heationg is required in the UK throughout the year, theoretically the direct heating-only use of electricity and/or gas etc, will be reduced.
And this device-sourced heat will effectively been recycled, ie used twice, firstly in its intended use for computing etc; and secondly for house-warming.
Where-as the direct heating-only consumption for central-heating will have only been used once.
Away back over half-a-century, we heated our electronic research labs by switching the equipment under test ON "first thing", as the central heating system was unable to bring the ambient temperature up to a reasonable level - so we both tested the equipment and created comfortable working conditions from the single source and "recycled" the enrgy, to use today's terminology.
As reported by another forum member, I wonder how much energy etc was taken up by scrapping four devices; and now using a fifth, apparently caused by frequent user-switching to save small amounts of energy?
And were any of the four failed devices stripped down and recycled in any way?