It maintains thermostat setting for days then stops working and will not come back on until we turn it off for 10 minutes and it comes back on and works for days again. We can't make it fail when the repair man is there.
What order from most to least likely would I check out what parts that could cause this?

Restricted air flow, Dirty flame sensor or burners is the starting point. That are some of the faults that do not require manual reset.