Why would my central heating unit run for one hour straight without cutting off if it has reached the programmed temperature?

It shouldn't. A couple of causes:

Thermostat shorted out (red-white wires)
Thermostat "fan" switch is on
Shorted wiring in the furnace itself


If the temperature continues to rise, I'd suspect something like #1.
If it runs, but is blowing cold air, I'd suspect something like #2.

Hope this points you in the right direction.
WallyH

Is this a new thermostat

Do you have a heatpump and its real cold outside? Questions, questions we all have questions. If you can answer this one and the previous it will start to narrow it down and get you back on track.

We also have this issue with our heat pump. Our house is 2 years old with the original thermostat (as far as I know.) It has been colder than the heat pump likes, but still, if the thermostat is set at X, and the temp gets to X, shouldn't it turn off? The fan is on auto, so that shouldn't be it.

My solution has been to switch to the electric heat at night, on our White-Rodgers thermostat it's called emergency heat. I know this isn't energy efficient, but otherwise, we're bugged all night hearing to the fan blow, which cannot be energy efficient either.
I do wonder if I'm hurting the system by doing this.

If your out side temps drop below 30 degrees your heat pump will run around the clock trying to keep up. If you loose 2 degrees on your set room temp the aux heat kicks in with the heat pump to keep you warm, when it starts to gain on room temp the aux heat drops out(system will run all the time with out catching up). Some times its better to switch to emergency heat at the thermostat to save money and cycle the equipment. I'm not saying nothing is wrong with your heat pump but they do run all the time with extreme temps.

Thanks, sounds like my instinct was right on not just wearing everything out by leaving the heat pump on when it's too cold.

I'm still curious about the continuous running of the heat pump when the thermostat shows that the house is heated to the set temp, while the emergency heat will turn off when set temp is reached.

Is this a common occurrence with heat pumps? If it's normal, fine, otherwise it seems the heat pump & thermostat aren't communicating properly. Would that maybe be a wiring issue?

I hope to really understand all this someday, sure wish the previous owners had left the manuals.