If a burning smell is coming from the freezer, this usually means the compressor or another electrical component has burned out. This can happen due to the compressor motor running too frequently. Depending the age of the refrigerator unit, it may need to be replaced.
The compressor is the driving force behind the circulation of air through the refrigeration tubes. The thermocouple of the refrigerator dictates when the compressor powers on and begins circulating air. If the thermocouple fails, or if the fridge is overloaded with items that prevent air from effectively circulating, it causes the compressor to work too hard and burn out.
When this process occurs in a freezer, it is possible for ice to build up around the coils. Therefore, it is necessary to defrost the freezer at regular intervals if it does not do so automatically. Also, the owner should avoid placing the refrigerator too close to a source of heat, and should periodically inspect the seals to make sure they are airtight.
Faulty wiring or loose plugs can also cause the compressor and other electrical parts to burn out. If the cord is frayed or if the plug does not stay seated in the outlet, the unit needs to be repaired or replaced immediately to avoid a fire hazard.