Circulation Fan
Freezers and refrigerators have two fans. The circulation or evaporator fan moves cool air from the freezer to the refrigerator through a vent in the shared wall. In stand-alone freezers, it circulates air through the freezer. If the fan's motor burns out, you will notice a smell that is much stronger when you open the freezer door than when it is closed. The smell might be stronger near the refrigerator's vent, and you won't hear the fan working at all.
Condenser Fan
The condenser fan is located under the appliance in most models. It keeps the condenser coils cool. If this fan stops working, the coils may overheat, causing a burning smell and causing the freezer to stop working. The fan motor itself may also burn out. In either case, you won't hear the fan working at all, or you may hear it trying but failing to start up.
Compressor Issues
The compressor unit consists of a pump and motor. This is the "engine" that runs the freezer, pumping refrigerant through the unit. If the motor burns out, you will hear it try but fail to start -- it may make a clicking noise, or run for a few seconds and then quit. Or, the start relay may be burnt out. In this case, the burnt plastic smell will come from the back of the unit rather than the inside.
Defrosting Problems
Freezers have self-defrosting units that consist of a heater, a defrost timer and a thermostat. Every six hours or so, the timer triggers the heater to warm up the evaporator coils, melting the ice and frost that builds up on them. Once the temperature becomes warm enough, the thermostat triggers the compressor to turn back on and begin the cooling process again. This cycle keeps you from having to manually chip ice and frost out of the inside of your freezer. If the heater burns out, you may notice a burnt plastic smell that is strongest inside the freezer, near the back panel. You will also notice ice building up on the inside walls of the freezer. If the thermostat or timer malfunctions, the defrost heater may continue to warm the evaporator coils indefinitely, creating a burnt plastic smell and causing the freezer to stop cooling.