Anytime the engine is OFF,you save gas.On a warmed up engine,if you shut off and then in a few minutes turn back on,it will not use alot of gas to warm back up,as say a cold engine.Still uses up fuel though,but minimal.
Alot of the old stories were true for older cars,Yet the new cars today run high pressure efi fuel systems that are pretty much well tuned so even when idling,they sip gas and don’t waste it like the old carburetor cars that would just keep flowing fuel into the intake manifold.
Yet when idling very long times,you waste gas.If you are delivering,keep the car running,unless your car is way out of tune and is burning gas at a high rate.The reason for this is starter life and battery.They take alot of power to turn the engine over.Turning a car on and off each time you stop can foul the plugs,wear our the starter,the solenoid,the ignition switch and or drain the battery some.
When you first start a cold engine,it has to idle higher and use more fuel until it warms up.It runs in open loop,until it warms up,becomes closed loop,and then relies on the oxygen sensor for adjustments.
Thats why some cars with vacuum leaks or efi problems around the intake or engine,run better while the car is cold (in open loop)but warming up,its running on a fixed program with extra fuel coming in.It is ignoring the 02 sensors until it reaches a desired temp.
It makes up for the problems until the 02 sensors take over (in closed loop)and then if there is a problem,it idles like garbage,shakes,stalls and wastes gas,because the 02 is getting some crazy readings across the board and trying to tell the cars computer how to compensate at the fuel system.