Is it necessary to shut off the water supply to a radiator in order to replace a radiator valve?
You need to drain the water from the heating system, effect the remove and replace, then refill the heating system. 

Some circulating-water heating systems are filled with tap water from the home's water supply, while others are isolated from the water supply plumbing and are filled with some other liquid. My experience is entirely with the most common tap water type, so I cannot tell you how to empty and refill an isolated system. 

If you have a tap water connected system, you need to find three connections on your heating system, one where the water is admitted, one where it can be drained, and one where air can bled from the system. 

The water will be admitted into your heating system from your home's cold water pipes, at a connection with a valve, pressure regulator, and probably a backflow preventer. Look for a pipe from the house plumbing into the furnace or into the pipes near the furnace. Look for a pipe interrupted by a valve and one or two other fittings. 

The water can be drained from your heating system via a faucet or similar valve near the bottom of your furnace. It will have threads on its spout to accept a garden hose. Attach a hose to this faucet and place the other end of the hose outside your house, at a point lower than the bottom of the furnace. 

The air bleed valve is usually found at the top of each radiator. These rarely have a handle attached; you are supposed to open it with a small spanner or a special key. 

To empty the heating system: turn the furnace off, close the admitting valve, open the drain faucet, and open the highest air bleed valve in the house. Wait until water stops flowing from the end of the hose. 

Use two spanners to remove the broken thermostat valve, and to install the new one. 

To refill the heating system: close the drain faucet, open the admitting valve, and wait by the air bleed valve with a pan or bucket until water comes out, then close the air bleed valve. Now visit every radiator in the house, open its air bleed valve, wait with your bucket to catch the water that comes out, and close the valve. 

If you have trouble confidently locating these places on your heating system, take some photos and post them here. Include a picture of one of your radiators, so you can get specific pointers from someone who has experience with your make and model.