I have an older Honeywell thermostat that is exhibiting this problem.

I will set the thermostat to around 72*F.
The function switch is set to HEAT (not COOL or OFF)
The fan is set to AUTO

Even if the thermostat temperature reading is less than 72*F, the heater will not kick on. I have not tried in COOL yet.

If I desire to, I can put the fan on ON and the system will begin heating the house, but this of course will continue to run until I put the switch back to AUTO.

Sometimes I can switch the function from OFF to HEAT a few times and slide the temperature up to 80*F and the heater will kick on in AUTO mode.

Is the thermostat simply not calibrated? Or is this a furnace problem?

The first thing you need to check is the thermostat. The fan switch doesn't control the heat, it just controls the function of the blower. When in the on position, the blower should run continuously. When in the auto position, the blower should start when there is a call for heat, instantaneously for heat pumps and electric systems, and usually upon temperature rise in a fossil fuel furnace. If you leave the blower switch in the auto position and move up the thermostat, will the blower eventually come on? Putting the blower switch in the on position should not turn on the heat, just the blower. Is your thermostat level? Is it filled with dirt to impede its movement? Is it improperly placed in the home where it can't read the temperature effectively? Unfortunately there can be a lot of reasons things aresn't working properly... and only trouble shooting will locate the problem.

Good luck

If it's a Round (T-87) thermostat-they're practically bulletproof, and it's likely the thermostat isn't the problem.

From your description, it sounds like you have a bad fan relay, a bad fan/limit combo, or a speed leg gone in the motor and the furnace is running, but shutting down on limit. Do the burners fire up initially after the call for heat but drop out 60-120 seconds after ignition?
Do you get no ignition at all?

What model furnace? Does it have a control board or separate controls?

A little bit more info is needed to properly troubleshoot this-sorry.