Yes a car will start when the transmission is out.
As long as your cars transmission is in park then the car should start but if the transmission is out it will not move.
If your car has a manual transmission and the car transmission won't shift out of park or neutral or something is wrong with the clutch then the car may not start.
But when you have an automatic transmission and it's in park then the car should still start even when the transmission is out.
Usually you'll notice signs that your transmission is about to go out on you such as taking longer to shift up, burnt transmission fluid smell etc.
If your RPM is going up but the transmission is not shifting on up and allowing your car to pick up speed then your transmission is likely going out or you have low transmission fluid level.
Always check the transmission fluid every 200 miles or so to be sure you have no leaks or loss of transmission fluid.