Not unless you want to risk damaging your engine. When the temperature light comes on, it means your engine is overheating. An engine should not overheat if the cooling system is properly filled and is working normally -- even during hot weather or stop-and-go driving. Sometimes abnormal driving conditions such as towing a heavy trailer during hot weather may overload the cooling system's capacity to control heat, but usually a temperature warning light means trouble.
Stop driving immediately, pull over to the side of the road, shut the engine off and wait for things to cool down.
WARNING: Do not attempt to open the radiator! The radiator contains hot water under high pressure. Opening the cap could allow steam and water to blow out and burn you. There's nothing to be accomplished by opening the cap while the engine is still hot, so let your vehicle sit and cool off.
After things have cooled down (wait at least half an hour or more), then you can open the radiator cap and check the coolant level if you think it might be low. Place a rag over the cap first, and slowly turn the cap until it stops at the first detent. This should allow residual pressure to be released. Wait until all pressure has been released before removing the cap the rest of the way.
Add water to the radiator if it is low. Actually, you should add a 50/50 mixture of antifreeze and water to maintain the proper freezing, boilover and corrosion protection. But in most cases when this happens, it's an emergency situation and you don't have a jug of antifreeze handy. If that's the case, you'll have to add antifreeze later.
Visually inspect the system for leaks. Common leak points include radiator and heater hoses, the water pump and the radiator. Internal leaks (such as a crack in the head or engine) can't be seen and can only be diagnosed by pressure testing the cooling system.
See related question #14 for other causes of overheating.
If you keep driving an engine that is overheating, it can cause serious damage. The engine may start to knock (caused by detonation), which in turn can cause piston, ring and head gasket damage. As the internal parts of the engine expand from the excessive heat, clearances may be reduced to the point where metal-to-metal contact occurs. Valve stems may gall and stick, and the pistons and camshaft might scuff or seize.
Severe overheating also creates tremendous thermal stress in the head(s) and block. This, in turn, may lead to cracking and/or warpage. This is a common problem with overhead cam engines that have aluminum heads.
The only vehicle that can be "safely" driven when the temperature warning light is on a late model Cadillac with a Northstar V8 engine. Cadillac engineers designed the engine control system to automatically deactivate half of the engine's cylinders if it senses an overheating problem. This reduces the heat (and power) being generated by the engine, and it allows the "dead" cylinders to pump air through the engine for internal cooling. The system is designed to provide a "limp-in" mode so the vehicle can be driven to the nearest service facility or Cadillac dealer for repairs. It is not designed for continuous driving.