The compressor start device, also known as a start relay or start relay and overload kit, may have failed. This will usually cause the compressor to not start. If the burning smell is coming from the compresser area check the start device. If the start device smells burnt, or the compressor is trying to start but can't, the start device is likely bad and will need to be replaced.
Inspect the power cord prongs and wall receptacle for burn marks. The cord should plug tightly into the receptacle. A power cord that is hot is a sign of a bad connection. If the power cord is burnt or damaged it will need to be replaced. If the cord is loose in the receptacle, the receptacle will need to be replaced by a licensed electrician.
04 - Kenmore Elite Refrigerator Condenser Fan Motor
If the burning smell appears to be coming from the exterior of the refrigerator, it's possible the condenser fan motor is wearing out or overheating. You should inspect the component for any burn marks, especially near the wire connection points, and use a multimeter to test the motor for electrical continuity to help determine if the component has burned out.