On a gas or propane furnace, expect checks of the igniter, flame sensor, inducer motor, pressure switch, and venting. On an oil furnace, the technician typically replaces the nozzle and filter, adjusts the burner, and runs a combustion test, then checks the heat exchanger for cracks.
If replacement makes sense, the permit follows the fuel: a gas permit from the Building Department, which takes applications through its online portal, or an oil burner or LP gas permit from the Fire Department, whose fees are paid before an inspection is scheduled.
A furnace swap is also a good time to check the house's CO detectors against the Fire Department's rules for your home's permit date, especially if you expect to sell in the next few years.