Expect the technician to identify the system type, then check combustion, the flue and chimney connection, water level or pressure, the expansion tank, the circulator, and safety controls. On a radiator system they may bleed air, replace vents, or check pitch on steam piping.
If replacement is recommended, the permit depends on fuel: gas boilers fall under Inspectional Services at 525 Canton Avenue, which enforces the state plumbing and gas code, and oil burners and tanks go through the Fire Department on form FP-056. Ask whether the old chimney needs a liner for the new boiler.
Getting a heavy cast iron boiler out of a prewar basement can take time and extra hands. Clear the stairs and path, and ask the contractor how they plan to remove the old unit.