The correct answer is B) actuated by a boiler safety trip. According to the regulations governing the US Coast Guard Captain's License Examinations, all oil-fired main propulsion burners with automatic safety control systems must automatically close the burner valve when actuated by a boiler safety trip. This is a safety requirement to immediately shut off the fuel supply in the event of a boiler emergency or malfunction that could lead to a dangerous situation. The other answer choices are incorrect because: A) the burner being properly seated is not a condition that would automatically close the burner valve; C) the "trial for ignition" phase is when the burner valve is opened to allow fuel flow, not when it should be closed; and D) the flame in the boiler furnace being confirmed is not a trigger for automatically closing the burner valve.