I tried to do this with the door handle button and the doors won't lock, just make a beeping noise.

Good thing I guess because sometimes the car is so quiet, I am afraid I'd forget the turn off the car. This would remind me when I try to lock the car.

But I was wondering if it is possible to do it thru the FOB, say if you needed to run back into the house to get something but didn't want to turn off the engine.

 

Pressing the LOCK button on the FOB won't lock the door while the engine is running..

 

No, you can't lock a running car with the key fob either. Rather than risk having the car stolen or joy-ridden, I'd recommend turning off the motor and locking the car.

 

do a quick search on the subject, there is lots of good discussion on a workaround.