And, you’ll most likely have to do it while it’s not working.
I’ve never seen a disconnecting means for the 24V thermostat circuit, and as such have to believe that the OP speaks of the 120V disconnect.
Pondering this further, the fact that the OP can shut it off, turn it back on and it runs suggests that it is a safety circuit problem, but not something such as cad cell eye, flame rollout, or DI fan verification. The only component which comes to mind in that scenario is the combination fan control/limit.
I’d love to see a schematic of this unit. 
Odd you should mention this, because I spent a few hours this week fixing my furnace and it was a problem with the fan limit control (luckily, the high limit worked perfectly). I agree that disconnecting the 24V source directly would be odd, but I wouldn’t rule it out as some sort of creative ‘field engineering’.
F*#$%&ing banner ads and now the hamsters ate my post!
Keeve, please post the make and model# of your furnace. If there is a manual for it available online, we may be able to make some better guesses as to your furnace problem. Without that info, we’re pretty much whistling in the dark, although I’m fairly certain it’s not the disconnect switch you replaced.
FWIW, my shot in the dark is that your system is tripping out on a fan failure, either the main fan or the draft inducer fan. It only happens when it is very cold because that is when the system runs the longest. It is certainly conceivable that a malfunction of this type would require manual intervention to reset.
Does your furnace have a flame sensor? Our furnace wasn’t communicating with the thermostat… it was going on and off, not staying on, etc. I cleaned off the flame sensor (a little wire that is visible in front of one of the jets) with an emory board and it works now.
Next, I’m getting me a tan jumpsuit with my name embroidered over the pocket. Maybe a little lightning bolt, too.
So under a certain set of circumstances, the furnace fails. Time for some creative Mickey Mousing, which will befuddle any future owner 
Install a time switch in series with the switch you’ve been flicking. Set time switch to be on all the time except for a short off period just after the coldest time at night. Or go for broke and have the time switch do a reset every hour, and set up a relay to bypass the time switch if the furnace motor is running.
Or take it a step further with a microcontroller. Have it read the thermostat; if it’s on, check to see if furnace is running. If not, cycle power. Put a counter in the power cycle routine to disable things after a couple of cycles close together. Perhaps have an input to a gas sensor too.
For those with out a sense of humour, the above is in jest. Unfortunately, I’m certain any repairer has come across any number of crazy, stupid, and/or dangerous bodges, sometimes from a previous “professional”. Personally, I’m bitter right now after spending time and money this past fortnight getting the house waterpump sorted after it’s installation 3 years ago by a twit :mad:
The make is “Delco Heat”, but I don’t see a model number anywhere. I suspect the problem might be in the Robertshaw “700 Series Gas Heating Control”.
I finally figured out how to get a free webpage at GeoCities. I took some pictures, and put them, togther with descriptions, here. I know my HTML skills are pretty basic, so please don’t anyone complain about the formatting.
Thank you all for the various suggestions!
Model and serial is going to be adjacent to the gas valve-once the service door is off, a data decal listing that and AGA and other compliance numbers is somewhere within view.
Yikes! That’s not a furnace! It’s a boiler. The domestic coil has been abandoned, as evidenced by the plumbing going to the land of lost left sox, and the water heater visible in the background. sigh
Square one. What was the last time the expansion tank was bled, the automatic feeder set, and so forth? When running, what is the steady state system pressure? Is the air scoop bleed cap free and clear? How many stories to the dwelling?
That pretty much sums up my reaction after seeing the pictures. My experience with residential boilers is virtually nil. I wouldn’t have thought of any of the nifty questions danceswithcats asked. Still, it is suspicious that a power cycle restores it for awhile. Two possibilities come to mind; 1.) Some component is failing, and killing the power for a moment allows it to reset and work again for a bit, or 2.) you are tripping a safety which requires manual intervention to be reset.
I think possibility 2 is the more likely one. If so, then you should definitely ensure that your T&P relief valve isn’t rusted shut before you continue resetting a safety control designed to keep the system offline until help arrives.
That’s all I can offer for help. Good luck, and listen to danceswithcats. He is the expert here.