Looking deeply into the past
 

I seem to remember an instructor when I did my ATPL way back discussing range / endurance when he was a nav on Bristol Brittanias.
Seemed a suggestion was made to BOAC that three engine holding or long range cruise gave dramatic decreases in fuel consumption with corresponding increase in range / endurance. Don't Nimrods and Orions do similar (shut down two) in their various roles?
Seeing one engine stopped might spook a few passengers, but that's never bothered anybody before!!
Seem to remember something about a concept business jet with two engines of differing power output designe dto have both running for T/o or landing but only one operating in cruise.
Just some ramblings from a dinosaur.

yarrayarra points to some valid issues. Turbine engines tend to be less efficient at lower power settings (ie thrust falls off faster than fuel burn).

Thus shutting down an engine or two, and running the others at higher power, can definitely save fuel - especially if you are not in a hurry. It's been done for decades on the P-3 Orion when loitering or low-speed patrolling.

Of course the Nimrod and Orion are unusual, compared to airliners, in the need to be able to conduct prolonged low level cruising.