My heater only turns on if the temperature inside the home is 68 degrees or above. If the temperature inside the home is below 68 degrees, I can hear the thermostat click when I turn on the heater but I get no hot air until inside temperature is above 68 degrees. Once the inside temperature reaches 68 degrees due to daylight or whatever, then the heater turns on and works without a problem and maintains whatever temperature I set as long as it's above 68 degrees. What could possibly be the issue?
It looks like your thermostat is wired wrong way: call for heat wire is connected to call for cool (probably at furnace side, as you said replacing the stat was unhelpful).
So if temp is above the one you set the heating starts as it 'thinks' it have to cool.
Please post a picture of wiring both sides (furnaces and thermostat)
Seems really odd. I'd try to replace the thermostat with a cheap one to see if it resolves the issue. If it does your thermostat is buggered up. There are ways to bypass the thermostat to check it too but you need to be a little handy. Comment back if you want to know more about testing your thermostat.