Oil Furnace And Blower Turns On By Itself (Even With No Thermostat Connected)

I attached an image of the furnace which displays the model number and other information on the tags. The model numbers I can find are: Beckett AFG Oil Burner Honeywell RTH6580WF1001/W Thermostat Honeywell R7284B1024 Electronic Oil Primary

My oil furnace will run normal most days doing the following:

1. We wake up and turn the thermostat up to 72F
2. The furnace heats the house as expected to 72F
3. The house will drop a few degrees and the furnace will kick on to get it back to 72F

This happens every few days, sometimes multiple times in one day:

1. After completing step 3 a few time, the furnace will turn on by itself (blower and heat/flame)
2. When this happens the temperature in the house will already be 72F (or possibly 73F)
3. It will run for 1-2 minutes, then all power to the furnace and thermostat will be lost (just like if I turn off the emergency switch)
4. It will continue to happen even if I turn the thermostat to the off position OR remove the thermostat from the wall
5. We have an electric exhaust fan, so when the power is lost the fan does not continue to spin and exhaust fumes build in the basement (slowly though since it is only running for a couple minutes, but sometimes we don't notice the issue for hours)

To solve the problem:

1. I turn off the emergency switch and let the unit sit for a few hours
2. I turn the switch back on and everything will work properly for hours or days

Other notes:

1. The unit never goes into lockout
2. Everything has been preffesionally serviced/cleaned at the end of the last heating season
3. I have had 3 techs from 3 different companies look at it and they are not sure what the problem is (although none of them have tested any parts or been in the house while the problem is happening)
4. I have forced hot air with air conditioning
5. The furnace has a 24V transformer and my thermostat requires 24V
6. Last year we were having a lockout issue, the first tech changed the pump and then the second tech changed the primary (changing the primary resolved the issue and the oil furnace ran perfect for the rest of the season)
7. We have an electric water heater, so the furnace is only used to heat the house