Thermostat failing?
 

Furnace at the house has been working sporadically. Sometimes, it kicks on, the gas flows and lights, but the fan does not kick on and circulate warm air. Sometimes it will work fine. Tried reset button, pulling batteries, setting the desired temp higher and lower to get it to come on, but it only works occasionally. Not a new thermostat, but not old enough that it has a mercury switch in it.

Is there a way to test it? Replacing it is something I think I can handle myself, but don't want to if it is not worthwhile. Is there a switch in the furnace that may have failed?

Modern furnaces have circuit boards and are "smart" and it is possible the circuit board has gone out.

Is it a gas furnace?

I will give you my 30 year old analog heater experience.

If it is a gas furnace, and presuming the installer followed standard common sense thermostat wiring.

If you can get the top off the thermostat (may be two screws or may just snap off) you will see the wall plate and the terminals. Red should be common (24 VAC) and white should be heat. Short those two terminals and the heater should come on.

This does not apply if you have a heat pump.

Had a 22yo Lennox PulseAire that would do something like that:Start stop start sputter maybe kick on after a while.
It was the flame sensor.
If the computer doesn't see heat it won't keep the gas on.
Guy cleaned it and it would fire up fast after that.

It is highly unlikely a mercury switch thermostat is not working.

It could be what John said or from the symptoms you have described either the controller has failed, or the low temp switch is not telling the controller to start the fan. Then the fan does not start and you go into either a time-out mode or overtemp.

it is also possible the thermostat turns the furnace off before the fan can start, but that's unlikely.

the fan motor probably has a capacitor. that could be going bad.
you really need to wait until it is not running and check for power at the motor.
you may need to disconnect the wires to verify power.

to check the capacitor remove the wire(s) that goto the motor, then apply power to the cap. if there is NO power on the other terminal the cap is bad.
you have to remove the wires, power can feed back thru the motor to the other side of the cap and it will look like it is good, or just replace it. they are very cheap.

mine was doing the same thing, sometimes it would not run, bad fan motor.

Flame sensor. Google to see what it looks like and where it is. Then clean with fine sand paper.

Magnets.

It's the magnets.

Man, I love this place
Flame sensor cleaning procedure, looks like even I could do it.

Says to use a crisp new bill in the comments, as the sandpaper could damage it.

A gas furnace will have a pressure switch with a vacuum line looking hose between it and the fan housing. It's pretty common to have the outlet of this hose at the fan housing get clogged with lint or the sorts. So take the vacuum line off of the nipple at the fan housing and using a small stiff piece of wire/toothpick/pipe cleaner/small allen wrench poke it in the nipple a few times then hook the vacuum line back up and see if she works.

We cooked the mother board in our 15 y.o. furnace. Cost about $150 as I recall. There are a bunch of sensors, relays and other mayhem in any "modern" furnace that could cause your symptoms. There is a "trouble code" that is translatable with a chart inside the door of our furnace that tells you which system is non-op - it's a series of flashing led lights.
Since yours is lighting it's not likely the "hot surface ignitor" - another part that dies - usually on the coldest Sunday night of the year... I keep a spare on hand anymore.

Just my two cents:
As I said if you can get to the base plate of the stat jump R (red=hot) and W (white=heat) and if the heater comes on you need a new stat.
Back when I did HVAC repair everything was analog, several safeties in a chain, if all safeties were go, click, heater on.
Now everything is digital.
Long story short unless you have time to burn if it is the heater call a repairman, he probably has parts on the truck and can swap out a circuit board to get the heater up and running.

Oh, and if you screw up the heater it can be a life and death thing if something goes terribly wrong.

If your furnace is a HE with a condensate line...check that to see if it is obstructed.
Mine was once, and it would do the same...start, run for a few seconds and kick out.
Dried slug in the line...cleaned it out, worked fine.

The the gas valve should not turn on until the fan starts and the pressure differential is confirmed. His furnace is getting past that check....

One of our AC units was cycling on and off sporadically during the summer - similar to what you report. It would cycle on only for a sec or two, then off, then within 5 or 10 minutes, same thing again. The AC technician diagnosed it as either the control board or the thermostat. I swapped the thermostat with one off another AC zone in the house and the problem persisted. I was able to save about $250 by finding an OEM control board off ebay rather than purchase through the mfr.

Not knowing the model of gas furnace does it have a circuit board? Some have led's that will flash a code. If you do there will be a small plastic window the size of a quarter on the fan door. Try and count the flashes. Sometimes the codes are listed inside the door.
If there isn't a circuit board dis-regard what is above.
If the flame sensor has a white coating on the rod portion then it could use a cleaning. If you are going to clean it turn the power off to the furnace. Sometimes there is stray voltage at the sensor, just enough power to give you a slight jolt. Ask me how I know.
One other thing, if you turn the fan switch from auto to on and the fan comes on then the motor is o/k.
Have you checked the filter?
I hope this helps.

A couple of years ago, my furnace went on the fritz. I called the furnace guy and waited a couple of days. He showed up and, as we walked downstairs to the furnace I told him the symptoms. He opened the panel, reached in and yanked out a 5" vacuum hose. Took a drill and unplugged the aperture at the metal piece it had been connected to. Put a new hose on and voila. He suggested I have him service it while he was there. He spent a half hour or so and charged me $125 altogether.

Get a furnace guy to look at it. They do this kind of thing.

Most furnaces have LED lights in them. A Simple self diagnosis is to turn the furnace on, raise the thermostat with the door open, you may need to push in the door switch. The LED light will flash a number of times. Take that number and look on the back of the door and it will show you why the furnace isn't running correctly.

Good Luck

If it's lighting but the fan isn't coming on, it's not the thermostat. If it's lighting and the flame stays lit but the blower doesn't come on, it's not the flame sensor. If you have a modern furnace with a circuit board, I'd say your fan relay on the board is going out. If you want to test you can turn your fan switch on the thermostat to the ON position. This should energize the fan relay and bring on the main blower.
As a note, if you ever question a thermostat failing, make a small jumper wire, remove the wires going to the R&W terminals on the circuit board and put the jumper on the R&W terminals. This will signal a constant call for heat and the furnace will run until you pull that jumper.

Edited due to incorrect information regarding limit switch.

Sequence of events
1) thermostat calls for heat and sends signal to furnace to start draft inducer.
2) Draft inducer starts and pressure switch closes turning on the hot surface igniter.
3) Once igniter is glowing, gas vale opens and main burners kick on.
4) When the heat exchanger reaches a certain temperature, clixon switch closes and starts blower motor.
5) When thermostat is satisfied, burners turn off but blower motor continues to run until temp in heat exchanger compartment is low.
As others have stated, could be a bad relay on the control board or a bad clixon switch

Flame sensor was the answer, or so it would seem. Naturally it went out when the temps outside are in the low 30's