No hot surface igniter

If you have a switch that can manually turn on the fan, try it to verify the fan works. This is what I think is happening. The normal operation should be thermostat calls for heat, inducer fan comes on, pressure switch closes, igniter glows, main burner ignites and then in 30 or so seconds the fan comes on and stays on until the thermostat is satisfied. Once the thermostat is satisfied the burner shuts off and the fan runs for approx 120 seconds then shuts off. I think the control board that automatically turns your fan on is faulty. On most modern furnaces if the auto fan control fails and the furnace gets to hot and limits, when the limit opens and shuts off the burner it also energizes the fan motor to cool the heat exchanger.

For this answer, we need a model and serial number.
Generally, be sure the power is on, does the heat moving device run? (Fan for air, or pumps for water)

The hot surface igniter will not work if it is cracked , you can get replacement parts which are easy to change but do not put your fingers on the surface of it ,

ignition failure. insure air pressure switch closes, hot surface igniter is getting 120volts. check continuity on hot surface igniter, make sure all gas shut off valves are open and gas valve is turned on. make sure there is no blockage in the exhaust

main fan should come on once furnace has heated up to supply hot air to the house. make sure proper voltage going to the fan. fan relay in board may be shot

check continuity on hot surface igniter. no measurable ohms reading = bad igniter. also insure 120v getting to the igniter

If this is a hot surface ignition system, the sensor may also be your igniter.

Sounds like either the flame sensor is not working.
The sequence for starting the furnace is:
1. extraction fan on
2. pressure sensor (for fan)
3. ignitor (spark or glow plug) comes on
4. gas valve opens up
5. ignition of flame
6. flame sensor (or temperature sensor)
7. plenum temp. sensor
8. main fan starts
It is most likely that the flame sensor is dirty or gone,
or (and I hope not), the module is gone and needs to
be replaced.

The entire igniter should glow, if it does not the temp to ignite the gas is not met and the burners will never light. When replacing the igniter never touch the element part becuae it will create a hot spot on the igniter and fail.

the fan you hear come on at a call for heat is most likely just the inducer motor. the click you hear is probably the relay for the hot surface ignitor which should glow red, then in turn ignite the main burners. your going to have to determine if the ignitor is coming on. if it isnt the ignitor is bad and will need to be replaced. if it comes on and than shuts off after a few seconds the ifc board is probably at fault