Expect the technician to start by asking about your fuel and what happened just before the heat stopped. On an oil system they'll typically check fuel delivery, the filter and nozzle, ignition, and the flame sensor. On gas, they look at the igniter, flame sensing, venting, and pressure switches, and on any boiler the water pressure and circulator.
Many fixes are a part swap. If the boiler, furnace, or oil tank has to be replaced, though, permits come into play: Inspectional Services at 525 Canton Avenue covers gas, sheet metal, and electrical work, and the Fire Department covers oil burners and tanks. An emergency doesn't remove those requirements.
Older Milton houses can mean tight basement stairs, low ceilings, and fieldstone or block walls, so clear a path to the equipment and the tank before the contractor arrives.