A technician typically watches a full heating cycle, then checks the igniter, flame sensor, gas pressure, inducer, and pressure switch, and measures temperature rise across the furnace. A combustion and CO check around the appliance and its vent is a sensible part of any furnace call.
In older Rockland homes, ask the technician to look at the ducts while they're there: duct joints, return air paths, and any runs through an unheated crawl space or attic. Sometimes the fix for a cold room is duct sealing, not a new furnace.
If the furnace needs replacing, gas piping work goes through the Building Department's gas inspector, and an oil furnace replacement or tank removal goes through the Fire Department. The Building Department doesn't schedule inspections on Fridays, so finish dates usually land earlier in the week.