On a repair call, expect the technician to check the filter, igniter, flame sensor, pressure switch, and blower, and to test for gas leaks or carbon monoxide at the flue. On oil furnaces they'll look at the nozzle, filter, and electrodes, and many will suggest an annual cleaning.
For a replacement, the permit path depends on fuel. Avon's plumbing and gas inspector handles gas furnace permits, and the Avon Fire Department handles installing or altering oil-fired equipment. The Building Commissioner enforces the state building code alongside both, and every application goes through PermitEyes.
If you're switching from oil to gas, the job also covers a new gas line, venting changes, and often removing the old tank, which needs its own fire department permit. Ask for one contractor to coordinate all of it.