On an oil system, which is what most Halifax technicians will find, a no-heat visit usually starts at the tank and filter and moves on to the pump, nozzle, ignition, and the burner's safety control. On gas or propane equipment, the technician checks the ignition sequence, flame sensing, venting, and any error codes. Expect a combustion check once it's running again.
Swapping a part normally doesn't need a permit, but replacing an oil burner or tank does. The Halifax Fire Department issues those permits on the state's Form 1, while gas appliance work runs through the gas and plumbing inspector at Town Hall. A technician who finds a cracked heat exchanger or boiler section may recommend leaving the system off until it's replaced.
If the storm that caused your outage also put water in the basement, say so on the call. Equipment that sat in water, whether near the Winnetuxet River floodplain or anywhere else in town, should be checked before it is restarted.