Check the high limit switch. If the furnace is allowed to continuously over heat, It's likely the high limit switch has burned out.
Make sure that the outlet vents are clean, and clear of debris. You need good air flow both in, and out of the system. also make sure the heat exchanger is clean, so it can transfer heat properly to the passing air.
If the thermostat has an option for Fan, switch it to Fan and make sure the fan starts. If the fan does not turn on, you should contact a trained HVAC tech. Capacitors can be deadly if not handled properly. You could accidentally discharge the capacitor into yourself, leading to electrocution. Or short the capacitor, leading to an exploding capacitor. In either case, it could be deadly.
If the high limit switch was bad, the furnace should not fire. When the high limit switch trips, the flame/heating element will be shut off and the blower will continue to run to clear the heat. The blower will continue to run until either the system is reset, or the limit switch closes.
I believe high limit switches are normally closed, and they should always fail open. Which means when they go bad, or the temperature gets too high they open the circuit. To test a limit switch, you'll want to check continuity to determine what state the switch is in.
- If there is a serviceman switch, turn it off. If not, shut the power off at the breaker panel.
- Open the access panel and locate the high limit switch. It should be somewhere on the hot box.
- Disconnect the wires from the switch (don't forget which goes where).
- Using a multimeter, test for continuity between the leads on the switch.
There is an easier test to check limit switches, but it should only be performed by HVAC techs. So I will not discuss it here.
If the circuit is open, you'll want to replace the limit switch.
If the limit switch is adjustable, make sure it is set properly according to the manufactures specifications.
If the limit switch tests OK; or you've replaced it and still have the issue, you'll want to focus on the thermostat.
From your description, it sounds more like a problem with the thermostat. Most safety mechanisms in furnaces do not let the furnace fire if there's a fault, so if you're getting flame I would not suspect a limit switch. I would suspect either the thermostat has gone bad, or it is miswired. If the fan is always on I would suspect either the fan switch has failed closed (is always in the ON position), or the fan wire was incorrectly bonded to the voltage feed line.
Thermostats are basically switch boxes. They'll have one live wire [R] (usually 24v in the US, or line voltage in other parts of the world), one heat call wire [W], one blower fan call wire [G], and optionally a cool call [Y], and/or C wire. When the thermostat wants to warm up, it will connect the R to W and G which tells the furnace the thermostat is cold and needs heat. If it's hot it connects the R to Y and G, to tell the furnace it's hot and wants to be cooled down. If you have a FAN control and you turn it to ON, the thermostat connects R to G and the blower fan turns on.
The easiest way to rule out the thermostat, is to replace it. Troubleshooting thermostats is a pain in the ass, so it's much easier to simply switch it out.
If you've changed the thermostat and you're still having the problem, call an HVAC tech.
Best Answer
Furnace blower fans will continue to run for a bit after the flame shuts off to strip the remaining heat from the heat exchanger and bring it into the rooms the furnace is heating.
Think of it this way - the gas valve (the flame) is controlled by the thermostat, but the blower is controlled by a sensor mounted in the heat exchanger. If the heat exchanger temperature falls within a certain range the blower will run to make use of that heat. You will also notice that the flame will run on its own upon startup without the blower for a short period until the heat exchanger warms up enough to justify running the blower over it and delivering heat.