Remote start is not to blame here. A manufacturer installed remote start system will shut the car off if a door opens. And the car should also shut off automatically after 5 to 10 minutes.
Unfortunately, after market remote starter does not offer this capability, so with the new trend of monthly charges, if you like certain brands like Toyota, and you want that type of secure remote start, you have to pay $20 per month or more for the life of the car.