If your furnace smells like burning plastic you need to have it serviced immediately. Normally when a furnace starts smelling like an electrical fire it really is experiencing an electrical fire, or at least the first stages of one occurring.
The electrical wire burning smell (plastic burning) coming from your furnace is nearly always the blower motor windings becoming too hot or the transformer burning out. If it is the transformer burning out your motor should turn on and stay on until you unplug the furnace and you will have no heat or cooling. At this juncture, if it is the furnace motor, the motor has already been damaged and is living on borrowed time. Replace it as soon as possible. There are a number of reasons for these types of furnace breakdowns
Most new furnaces are equipped with a fuse relay to prevent the transformer and the circuit board from burning up. Many times a homeowner will pick up a thermostat at the local big box retailer and switch out the thermostat either mis-wiring it (Green is not ground in 24v systems) or simply changing it out without unplugging the furnace. This causes a short circuit and within a second or so the transformer, circuit board and/or the fuse fails. Yes, I have burned out a few those transformers myself over the last couple of decades of performing heating and air conditioning repair.
If the condensation line for the air conditioning evaporator coil becomes plugged up or the vent cap on the roof is not present the water can short out the controls in the furnace.
Water Damage: The condensation line become plugged up and causes water to fall down on the motor filling up the blower assembly. The motor will burn up when energized.
Too Small a Return Air Ducting: When the ducting is too small leading up to the furnace the blower wheel will not properly load up with air. It can operate in too large a vacuum. This allows the motor to convert the extra electricity it is designed to use into heat and the motor can burn up.
Debris in the Blower Wheel: If no filter is used tape, paper and other debris can become lodged in the blower wheel throwing it out of the balance and stressing the motor. If too much buildup occurs in the blower wheel due to lack of maintenance the wheel becomes heavy stressing the motor and eventually it will burn up causing that awful plastic burning smell.
Operating the System With the Blower Door Off: The blower motor and wheel are design with a close tolerance in loading conditions for maximum energy savings. If you remove the blower door and bypass the safety turn off (many times I have seen this) you can operate the system without the blower door on. This is a very dangerous situation as a vacuum can actually pull the flames right out of the burner compartment in your furnace. Removing the blower door allows too much air into the furnace and not enough vacuum for the motor to operate within its parameters. This increases the load on the motor and can cause it to burn out.
If the capacitor fails for any reason the motor will consume more electricity (amperage) than it is designed to use. This will increase the heat in the motor and cause failure.
Furnace blower motors will not last forever. They do wear out, bearings become tired and windings age. In general the blower motor in your furnace should last longer than 15 years under normal operating conditions.
If the blower wheel fails by throwing a weight or a rivet breaks it will become out of balance and this will cause strain on the motor causing it to wear out prematurely.