Not sure how old the furnace is. Probably ancient. Likely 20+ years old?
Seems to be model 01602/4-125. Lennox, oil, efficiency 80+
(I am rental tenant.)

I have three main questions below.

The current issue is that the (brand new 5 days ago) thermostat clicks appropriately and turns the heat on. Clicks again to stop calling for heat when the set point is reached, but the heat keeps coming. I have to turn it off with the main electrical switch.

I'll spare you the blow by blow of the last week (available if you wish to know). Generally, last weekend the heat never turned off while I was out to dinner, was pushing 83 when set to 74, and I had to shut it off by fliipping the electricity off. Over several days, the owner replaced the thermostat (now a 5-2 programmable Honeywell, being used without any programming while trying to get the heat working properly); changed the filter; replaced the ignition unit (he said it was sparking, I think); in desperation replaced the motor (very nice: much quieter during the on/off/on/off/we-don't-need-no-steenkin-hysteresis-setting-in-the-thermostat times). (if that would even govern; what do I know?)

Operation was unpredictable. At some points in this saga, the problem was that it wouldn't switch on. Later, it would cycle on/off/on/off until it had a nervous breakdown and needed to be reset to work at all. (It's in a basement, accessible only by going outside and down through bilco doors. In the snow. Have pity on me, HVAC geniuses!). Sometimes it would come on and stay on and hit the 80s again.

Finally, the owner opened up the control unit, and we could watch it fail. (Supposedly new in the spring so not immediately suspect; turns out to have been refurbished.) So yesterday, new control unit. And working! Yesterday, last night, this morning: warm house, no resets. Happy, happy, joy, joy.

This afternoon while absorbed in reading email, I failed to notice it was staying on. Hit 80s again. Turned the electricity off.

So, my questions: (1) What could make it know how to respect the call for heat but not respect the call to stop heating? And (2) Is there anything that I'm doing that could be damaging the control unit? And (3) How is the 'off' switch at the thermostat supposed to work, 'cause it doesn't seem to; is that significant info?

Okay; 3.5 questions.
Educate me some more, please.

Since the thermostat has been replaced & the overheating continues, I would suggest replacing the thermostat wire. I've seen cases of damage to the thermostat wiring which would cause intermittent shorts & overheating.
I've also seen the control contacts stick. This is easy to diagnose IF you can catch it in the act. One simply has to disconnect the thermostat wires at the control. If the burner stays on, the control is sticking. Replace control.

Thank you, Grady.

I had always thought it was an issue with the wires from the thermostat. Owner came by this afternoon, prepared to pull a new set. Instead, he found exactly where one had gotten close to an exhaust pipe and was melted. I believe the case is now closed. But your voice of experience was helpful support along the way.
Much appreciated.

Hope that was the problem & your overheating issue is a thing of the past. Should anything else crop up, I'll be here & do my best to help.