The technician will usually start at the burner, read any lockout codes, and test the ignition, flame sensor, and safety controls. On an oil system that ran dry, expect a filter change and air purged from the fuel line. On propane, a system that ran empty may need a leak check before it's relit, so ask your supplier about that step.
A repair to existing equipment is a different job from a replacement, which brings permits. Gas and propane appliances need a gas permit from the Building Department, whose inspector visits on Tuesday and Thursday evenings, and oil equipment can involve the Fire Department. If a replacement is needed, ask whether the contractor can restore temporary heat while the paperwork moves.
Think about access before anyone arrives. For a seasonal house, arrange for someone to let the technician in, and clear a path to the basement, the burner, and the tank.