There are two answers to this question:
Theoretically, radiators are completely optional on engines. Air cooled engines have existed for a very long time. The function of the radiator on these engines is replaced by cooling fins to extract the heat and release it into the air, and they also have oil coolers as well to carry out the heat. However, if your car was designed to require a radiator, you really do need one. There is no benefit to removing it.
That being said, you CAN run a water cooled car without a radiator, but it is risky and requires you to shut down the car whenever it gets above normal operating temperature and let it cool off via convection.
If you have a big container of water (like a cooler with an electric pump in the trunk or passenger area) you can pump water into the engine and instead of it flowing out into a radiator, you can pump that water back in to the cooler. It will eventually become boiling hot, at which point you must stop operating the car and either wait for it to become cool again or empty out the cooler and refill it from an external source of water like a hose.
If you have a big enough reservoir of coolant, you can cover pretty significant distances with this approach, but it requires that you keep the engine's water passages completely filled (so the coolant temperature sensor is covered in circulating water) and you have an accurate temperature sensor.
Even if you just seal off the coolant hoses and only operate the car with the water inside the engine, it will run safely until it becomes hot. At this point, you either shut it off and wait for it to get cold again or you keep driving and kill the engine. If you keep driving, your engine will be in serious trouble. The heads will warp, the head gaskets will blow and the oil will probably lose some of it's lubricating qualities. Whether the bearings coke up or the heads crack first is really dependent on the design of the engine, but you really don't want to drive the car like this.
Generally speaking, the less coolant you have in the engine, the faster it will heat up. The block itself has little thermal inertia compared to the water and the heat output of even small engines is extremely high. Gasoline engines typically generate kilowatts of energy, a significant portion of which (about 2/3rds) is lost as heat. A good portion of that is dumped into the coolant. Also, if you use water instead of anti-freeze mix, the engine will take longer to heat up, as water has much higher specific heat.