Will reducing the electricity power in an aircraft save fuel?
Yes, in theory, but you would have to do a total energy analysis to decide if it's worth it.  On the surface of it, generators consume fuel not being made into thrust. Watts being made to power things is watts not going out the tailpipe. 

A good example of the potential benefit is the switch to LED cabin lighting. A 40w incandescent light is more or less a 35w heater that makes a bit of light.  LED lights drop the power consumption by 80-90%, although if you need heat in the same space, you have to replace the heat being made by the lights, so if you did an overall energy balance analysis in that case, you might find the actual saving is negligible for the period when you need to heat the cabin in cruise at altitude (because the reduction in heat from the lights has to be replaced by more bleed air demand).  

If you need cooling, it's the opposite because the heat from incandescent lights is working against you, and it's a huge benefit to get rid of the heat produced by the lights, which reduces cooling demand.

So overall, there is probably some fuel burn improvement over time switching to LEDs, very small, but something (say, 200 30w lights consuming 6kW, replaced by 200 5w LEDs consuming 1kW, a 6hp saving, maybe 4lbs/hr - not nothing, but, but as I said you'd have to take more bleed when heating, offsetting a lot of that improvement, and you'd probably have to spread it out over quite a while to get serious money out of it). 

Most airlines that switch to LEDs do it mostly for reliability reasons, where labor hours savings and parts costs make a stronger financial business case than just a few pounds of fuel spread out over time, although the fuel-saving does add to the total.

Another area would be advances in avionics that reduce power consumption (you can just imagine the drop in energy demands when avionics went from vacuum tubes to solid-state in the 50s).  Even there though, it's the same issue with heat that is not really wasted when the cabin requires more heating than cooling, that steals some of the benefits you thought you were getting.

Reducing electrical demand with efficiency improvements also helps with weight savings (smaller, lighter generators).

On the other hand.... the 787 blows all that out of the water by going to the other extreme, making all sorts of services electrical, requiring really massive generators. I'm not sure that there is a net energy benefit to using electricity to replace bleed for anti-icing for example, and I think the main benefit is reliability (bleed air systems are a massive maintenance and reliability hassle).