Do you have an alarm on the car? Most cars will not activate the alarm if the car is running. For example, with my 2007 TCH, if I try to activate the alarm with the FOB while the car is started, I don't get the "alarm activated HONK" to let me know the alarm is on.

I don't know anything about the NAH. It does not make sense that Nissan would have allowed the car to be left on and the smartkey taken away from the car without an audible warning like the TCH has. With the TCH, if you leave the car on and take the key out of range, the car beeps loudly to let you know you are messing up.