Expect the contractor to start with power, fuel, and the thermostat, then move to the burner, ignition, and safety controls. On a hot water system they'll check pressure and the circulator; on a furnace, the blower, filter, and flame sensor. Most no-heat calls end with a repair or a part order rather than a new system.
If the fix involves gas piping or replacing equipment, a permit from West Bridgewater's Building Department is usually required, and its Plumbing and Gas Division takes those applications online. When a repair can't wait for business hours, ask the contractor how and when they'll file. The permit card has to be posted at the house for the inspector's sign-off.
If the equipment is unsafe to run, the contractor may shut it off and tag it. That's a safety call, not a sales tactic, and you can ask for the reason in writing.