You probably have an inducer blower which is a sort of exhaust fan for flue
gas (aka the smoke) on higher efficiency furnaces. This will typically
shut off with the gas burner when the thermostat says enough already, or
may remain on a bit longer with a time delay to clear the gases out of the
heat exchanger and reduce corrosion.

You're saying you only hear the clicking when the gas burner shuts off?
Does it continue until the main blower shuts off or just briefly as the
inducer coasts to a halt? Could be dirt on it. Many inducer motors have a
small fan to cool the motor and this will be a fan you can probably see
when you take off the outer furnace door. The inducer blower itself is a
squirrel cage blower wheel in an enclosure drawing on the heat exchanger
and blowing out the flue pipe. Without periodic attention the cooling fan
and motor (and perhaps a finger guard over the fan depending on type) may
get very dusty dirty.

Observe while it does its thing and shuts down and see when and where the
clicking is occurring. If you do see a visible motor cooling fan you
can...with furnace power off or at least turned off at the thermostat...try
giving it a spin. It should spin freely on its bearings (the blower wheel
you can't see, the motor rotor, and the fan you can see should combine to a
little bit of mass that if free from obstructions and on good bearings
given a push will spin easily for good bit). On the other hand just
observing normal operation seeing how long it coasts when it switches off.

It's also possible that the clicking could be nothing more than furnace
components contracting after the burner shuts off and making various clicks
and pops on account of that and being old.