There are a number of things that can cause electric heaters to smoke or emit strange odors. The main culprit is dust and the buildup of these dust balls in on or around the inside or outside of the unit. The vents could have dust built up in them, or even the heater coils themselves if your heater is stored in a dry, arid environment all season.
There are ways to prevent this from happening, as well as ways to get the unit back to the way it was. Some people may find that the issue is not dusty at all, but instead has broken, burnt, or frayed wires, or wire casings in the unit that are getting too hot. Loose connections can also play a roll in strange electrical smoke or odors, and can generally be fixed with replacement parts or some simple spot soldering.
Check the System for Dust
The largest culprit of this kind of dysfunction in electric heaters is unavoidable dust build up. This can accumulate over long periods of time without use, such as summer storage. The dust is everywhere in the air and can build up quickly unless you are running a humidifier to remove it from your home. You can use canned air to blow out the vents and try to push any clumps out through the ports.
If you cannot get to the build up on the inside. You may need to open the case and clean it out by hand. Dust anything that is built up on the heater coils and temperature sensor. Be sure that all the wires are held away from not only the heater coils, but the temperature sensor as well to prevent inaccurate readings from electrical discharges. Be sure to check any fans in the unit as well.
Check the Internal Wiring
If the odor smells like plastic, then you will want to check the electric heater's internal wiring. Look for casings or insulation around wires that might be melted or out of place. If you see wires dangling close to the heater coil, you will want to use some duck tape and secure them off to the side to prevent further damage.
You can use electrical tape to repair damaged insulation and fuse the tape shut with a little heat from a lighter. Once you have the wires sealed and out of the way from the coils, you should be able to reassemble the unit and test it for functionality.
Loose Connections or Blown Circuits
The only other thing that is known to cause smoke and odor from electric heaters is loose connections or burnt out circuits. In this case, you will want to test the circuit board with a voltage meter, or visually look for charred components. If you do see what appears to be burnt parts or loose connections, then more repair will be needed.
Some heater manufacturers sell replacement parts for their heaters, and you will want to check to compare prices. If parts can be replaced, and you feel comfortable with soldering connections and the like, this could very well be the solution to your problem.