I did that, recently. My fob battery died, I unlocked the car with the key, opened the door and... the car alarm went off. I'm not sure what the designers were thinking.
You turn the alarm off by starting the car, because the ignition has an rfid-like close-range reader which only requires passive circuitry in the key. That's how you differentiate between a break-in and the legitimate owner.
My car has push start (like many new ones) & has no keyhole inside (it has one in door to open the door). Although it has a seat/slot for the whole key to go in, in case of low battery. I assume that will stop the alarm. :-|
My car is also push start and I have to hold the fob in front of the start button for a short while before turning on the car if the fob battery is out.