Regardless of what is entered in the FMC the outside conditions are the outside conditions that exist. The FMC is just another instrument to help us manage the flight - entering this altitude or another for A/I on or off is irrelevant. Just my opinion
As Framer said,VNAV profile will be more accurate with proper data entered in the descent page of the fmc,including TAI on OFF.
EXENG,you must be aware of the N2 increase with EAI ON ,which means your idle will be higher hence adescent should be started earlier...if you delay the descent your fuel burn will increase for a fixed CI.
Same without fmc,youd have to increase your speed to to increase rate of descent if you selected tAI On,,maybe not by much,,but all counts ,even QNH setting in the descent page.
By the way I never tell my F/O where we will start the APU because I will not insult the intelligence of my colleagues - they know what is the best option for the airfield
You should have SOPS for that,no?