Yes the car will still operate without the Key fob once the engine is running.

What happens if you drive off without key fob?

The answer depends on the situation. For example, you will have a very different experience if you lose your keyfob BEFORE you get in the car or afterward. Simply put, if you lose the keyfob that enables your push-button start before you try to drive in your car, then you simply won't be able to start your vehicle.

Do you need a key for keyless entry?

Keyless entry systems allow you to unlock and lock the doors to your vehicle without using a key. Most modern US vehicles are equipped with a basic keyless entry system that includes a short-range remote transmitter.