I've heard of people starting engines in a build without a Radiator hooked up, to see if it fires.
If you have an old engine and want to see if it will run:
It can absolutely be started without a radiator. You will not cause any damage as long as the engine does not overheat. If you don't run it long enough for the engine to get too hot, it's not an issue.
To give you an example of how it could be beneficial:
I used to own a 94 Camaro Z28. The engine in it was a Gen-II LT1 350. This engine has what's called an Optispark Distributor. The Opti sits on the front of the engine behind the water pump. If you replaced the distributor, you'd want to ensure it's going to run correct before you put the water pump back on and fill it up with coolant. So, you start it up and let it run for a few seconds. There's not enough heat build up to cause any issues, so ... no harm; no foul.
An engine is a very large thermal mass. When the engine is cold it takes time to warm up. If you only run the engine for 15 to 30 seconds from cold there should be no problem. Running the engine any longer than that may cause the engine to overheat.
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.
Only based on your question. The answer is yes. You can run an engine without radiator. You just need to find an alternative way of keeping it coolor you will end breaking it. It's not immediate, but once the engine warms too much it will break.
There are many engines that were develop specifically with this purpose in mind and only use the air flow to keep the temperature down.
The engines that I know are mostly used in scooters and cars. One notable case is the original Beetle.
Be weary of running your water pump with out fluid, that could cause damage. But as far as just the engine, it won't be an issue as long as it doesnt get hot and is only run for a short amount of time.
Some cars designed for drag race pulls fill the coolant cavities with a hard material and remove the cooling system. This allows higher boost pressure for short periods.
Also be warned without coolant the temp sensor may not work because it will be in contact with air and not liquid.
