On an oil system, expect the technician to check fuel supply, the filter, the nozzle, and the ignition and flame sensor before anything bigger. On gas and propane equipment, the checks shift to the igniter, flame sensor, pressure switch, and venting. Point out where the tank, the emergency switch, and the thermostat are, and clear a path through the basement or utility room.
Most repairs don't need a permit, but replacing equipment usually does. In Kingston, the Fire Department issues the residential permits for installing or removing an oil burner and for propane tanks, while gas fitting permits go through the plumbing and gas inspectors at Building, Inspections & Zoning. If a failed boiler has to be replaced in a hurry, ask the contractor to explain which of those offices is involved and whether a temporary fix can keep you warm until the inspection.
Storm damage changes the order of work. If flood water reached a basement burner, controls, or tank, the equipment should stay off until it has been inspected, even if it looks dry.