On an oil system, expect the technician to check the tank level, filter, nozzle, electrodes, and the burner's safety control, then test combustion before leaving it running. On gas or propane equipment, the check covers ignition, the flame sensor, venting, and any fault code the control board shows.
If a repair turns into a gas or propane appliance replacement, the permit goes through Plympton's Building Department at the Town House and the town's Gas and Plumbing Inspector does the inspection, which the inspector arranges by text or phone. The contractor, not you, files that paperwork.
If the system can't be fixed on the first visit, ask what temporary steps will keep the pipes safe and what a replacement would involve, so you aren't making a large decision in a cold house without the facts.