Regardless of model or style, a fridge's main purpose is to keep food cool and reduce food spoilage. Like nearly everything mechanical, fridges are prone to various malfunctions, a few of which are evidenced by a burning smell emanating from the unit. Luckily for those experiencing the smell, a few do-it-yourself remedies exist that can help solve the problem and negate having to call in a service repairman.
Unplug the unit from the wall and pull the fridge away from the wall a few feet -- just enough to allow you to get behind it. Go to the front of the fridge and use your fingers around the door. Feel to see if the rubber seal strips around the door are hot to the touch.
Replace all the strips around the door if you find they're hot. The sealant strips are responsible for keeping the metal warm to prevent "sweating", and when they become loose they have a tendency to heat up and produce a burning smell.
Get down on your hands and knees at the rear of the fridge and check the coolant fan for large amounts of dust or debris. The fan is responsible for cooling the fridge's internal parts; when the fan becomes obstructed and can't rotate properly, the motor may burn up. Remove any dust clogs or any other debris to see if the fan can move on its own. If not, you may need to replace the fan or, in extreme cases, the motor itself.
Use a screwdriver to remove the cover plate on the rear bottom of the unit, below the fan's location. You may need a torch to be able to get a close inspection of the fridge's condenser coils. They heat up while the fridge is running; if too much dust collects on the coils, a burning smell can be produced. Wipe away all the dust with a clean cloth dampened with cool water.