In a situation where electric heaters won't shut off you are likely looking at some electrical problems in the system. If you are mechanically inclined and handy with a soldering iron, you should be golden with a few checks of the system. It is good to have a voltage meter to check for lost power in the unit once you have the case open for inspection. Be sure you follow all safety precautions to avoid getting burns or electrical shock.
Check the System's Temperature Gauge
Inside electric heaters is a coil, or thermal temperature gauge sensor. If this is not functioning properly, it will not detect that the heater has reached the desired temperature and shut off. This should be the first thing to be checked and replaced. It should be located close to the heater coils, but not so close that it gets incorrect readings from the heater's outputs. Crossed wires can cause the gauge to not function properly as well. You will want to check all connections and be sure that they are getting proper contact. Each individual part of these units, like the fan, heater coil, and temperature sensor run on their own circuit. These functions generally come together at the circuit board, where they interact with one another to provide functionality.
Test the Circuits with a Voltage Meter
One thing you can do with malfunctioning electric heaters is check the circuits on the main board for proper voltage transfers. Use a voltage meter to read the positive and negative poles of all your electrical connections. If you find areas that lose current, then the components could be burnt out, which will not allow electricity to be regulated or flow properly. This can inhibit the unit's shut off mechanism once it reaches core temperature. You can check with your unit's manufacturer to see what type of replacement parts you can purchase for replacement yourself. If you find spots in the circuit board that are not responding properly to the electrical checks, then it is likely that either a regulator is fried, or the actual circuit path on the board itself is. Look for burn marks or scarring of the board's surface for blown connections or components.
Is the Temperature Guage Snapped or Broken?
The most common malfunction in electric heaters is that the temperature sensor simply wears out or breaks. These can be switched out as a replacement part on most units, and you can even integrate a sensor from another broken unit as a spare part. These devices are generally made with the same electrical input and output standards, making it easy to replace. You can check with the manufacturer to see if they sell replacement temperature sensors that will fix your problem. On older units, this sensor is a coil that looks like a spring, and if it is broken, it will not function properly. On spring sensors, you cannot simply repair this component. You will actually have to disconnect it and replace it with a functioning part.