A technician will usually check the igniter, flame sensor, burners, gas pressure, venting, the blower, and the filter, and look at the heat exchanger for cracks or rust. On an older unit they'll often explain whether the repair is worth it given the furnace's age and condition.
If the answer is replacement, expect the contractor to file a gas permit, and a sheet metal permit if ducts are changed, with Inspectional Services at 525 Canton Avenue, then schedule the inspection. Oil furnaces go through the Fire Department's oil burner permit on form FP-056 instead.
Milton's older basements can be tight, with low beams and narrow stairs, which affects how a new furnace gets in and how venting is routed. Ask whether a new high-efficiency unit will need a new vent through the wall and where it will exit.