The plumber will check the heat source first: the burner and controls on an oil or propane unit, the elements and thermostats on an electric one, or the coil and zone valve on a boiler-fed setup. They'll also look at the tank, relief valve, venting, and how much sediment has built up.
If replacement makes more sense than repair, expect a talk about sizing, fuel, and venting. Some homeowners consider switching to a heat pump water heater, and Mass Save, the statewide program that utilities including Eversource run, is the place to check current incentives before you decide.
Replacements usually require a permit through Rochester's Building Department, with the plumbing or gas inspector arranging the inspection. Oil-fired equipment also involves the Fire Department's oil burner permit, so ask who handles each piece.