Just curious. Do these cars have some sort of engine shut done feature that will shut off the engine if the key fob is not near or is a certain distance from the vehicle?

 

Not that I am aware of. Once the engine is running, it will continue to run with or without a key until it is shut off. Obviously, you won't be able to re-start it without the key afterwards.

 

Oh ok thanks.

 

99.99% sure it doesn't. Safety issue. Imagine engine turning off because your angry GF throws the key out the window why you're doing 100mph

 

Yes, this.
Lawrence

 

Correct, no similarly-equipped car will shut down if the key fob is removed from the car after it is started for obvious safety reasons. I had a real-life experience with this a few years ago.

After leaving a restaurant, one of my friends wanted to hear the engine in my then new C6 Z06. I tossed him the key fob and invited him to start the car, which he did. We proceeded to walk around the car for a few minutes before saying goodbye, and I jumped in to go home.

On my way home, I realized that I didn't have the fob and almost instantly realized that he had probably put the fob in his pocket after starting the car. The car ran normally with no warnings or issues. Since it was a 6MT, stalling the car with the clutch would have been a show-stopper, but otherwise there were no worries.

I had my second fob at home, called my friend, and he had indeed stuffed the other in his pocket, which he returned to me the next day.