If the only demand is coming from the thermostat, then any activity on the part of the furnace means it got the signal from the thermostat. If you have a combined system where you get heat and hot water from the same unit, then the hot water demand could cause it to turn on.

Easiest test is to make sure the furnace isn't doing anything, and then crank the thermostat all the way up.

If the furnace keeps doing nothing, it is either the thermostat or the furnace. If the furnace does something, it's the furnace that has the problem.

there is a faulty connection from the thermostat to the burner switch or something wrong within the stack control.

I'd call a professional for this one.

While you can clean the soot out of a stack control yourself, it's a nasty job and there is no guarantee that's the problem.

My burner definitely needs a cleaning, I let it lapse because the guy I usually have do it yearly retired last year.

my guess would be air in the fuel line. I used to have to bleed the fuel line after the furnace had not been run for awhile. There was a bleeder valve on the oil filter housing and until clear oil was coming out (it was milky if air or water was mixed in) the burner wouldn't start running. Then again, that was 35 years ago...

Is the system maintaining the temp you set at the thermostat? You didn't mention if it's cold in your house.

If you have a boiler, it may already be up to high limit when you turn up the thermostat and the clicking you hear is the relay in the aquastat closing which powers the circulating pump but not the burner assembly as the temp is already at high limit.

If the temp in the house is what you have the thermostat set at (within a degree or two), there may be nothing at all wrong with your heating system.