My car also beeps when it detects that the key has left the car. The engine keeps running, but you obviously cannot turn it on again once you turn it off.
I had it happen without me leaving the seat (e.g. my wife has the keys in her bag/pocket, I had been driving, and she gets off the car to unarm the home alarm). The car is turned on by pressing a button, not by turning the key.
That seems like a reasonable setup. I'm having difficulty imagining how that could be hacked into the blackmail situation described above, since the sure way to avoid the beep is to keep the fob in the car.
I had it happen without me leaving the seat (e.g. my wife has the keys in her bag/pocket, I had been driving, and she gets off the car to unarm the home alarm). The car is turned on by pressing a button, not by turning the key.