If you have a problem with your cooling fan it may be an electrical problem. You don't want to drive you car in this condition because it can run hot.
Problem with the temp sensor.
When it's running hot you should hear the cooling fan running, if not that is your problem. If the cooling fan does not operate check the fuses, cooling fan relay, cooling fan switch and cooling fan motor. One of them is at fault.
Make sure that the fuse and sensor for the cooling fan are working. If these are ok then look at the fan motor, it could be bad.
You may have to replace the cooling fan relay or the cooling fan itself, I had the same problem and I first replaced the relay and it helped but then the cooling fan went out and I repalced it and now everything is fine.
cooling fan
Either the fan is no good or the heat sensor in your radiator, that turns the fan on is not working!!
a fuse or a bad fan motor
Could be a bad cooling fan on your Processer or a bad power supply.
Check the wire cnx. also good to replace the temp sensor for the fan. Just bypass the sensor to see if the fan does work first.
If the engine cooling system fan, check the fuse/relay to see if the it has blown out. If not the fuse/relay, the cooling system temperature sensor switch may have failed. For he A/C system auxiliary cooling fan, the same trouble-shooting method. If you are referring to the cabin interior circulating air fan - check the fuse and/or the switch that controls the fan speed.
The cooling fan light on your dash refers to the engine compartment cooling fan on the passenger's side of the vehicle, not the radiator cooling fan. In my experience with my '85 Mr2, that fan doesn't do a helluva lot, cuz mine only works if I unplug the temp sensor (sensor's dead and I don't wanna replace it) and I've never had a problem with overheating.
You need to take your car into a service center as soon as possible. The fan is on constantly because there is a problem with your car. It could be your thermostat or cooling temperature sensor.