Furnace shutting off, requires frequent resetting
Blocked flue and/or insufficient combustion air inlets.

Charging $300 without at least giving you a complete understanding of the problem was pretty lame.

That looks like the flue rollout sensor. If that is what is tripping, then you might have a blocked flue. Verify that the damper above that opens fully when the boiler turns on. Does it? Also check the combustion vent outside and make sure there's no obstruction like a bird's nest. Turn the water heater to pilot, turn off the furnace, remove the flue cap outside and inspect it with a flashlight. The button is on the bottom of the rollout sensor I presume (I cannot really see it).

If the flue damper does not open correctly then disconnect the actuator and manually lock it in the open position. Doing that will hurt efficiency a little, but the boiler will run.

The other obvious thing to check is combustion air inlets. I see no outside air inlets in the photo. There needs to be significant airflow into the room from outside to supply combustion and flow out through the flue. If the room is sealed tightly from the outdoors then that is a problem.