From the latest Boeing 737NG FCTM:
Autothrottle Use
Autothrottle use is recommended during takeoff and climb in either automatic or
manual flight. During all other phases of flight, autothrottle use is recommended
only when the autopilot is engaged in CMD.
During engine out operations, Boeing recommends disconnecting the autothrottle
and keeping the throttle of the inoperative engine in the CLOSE position. This
helps the crew recognize the inoperative engine and reduces the number of
unanticipated thrust changes.
Note: The autothrottle logic on some airplanes allows the autothrottle to be
physically engaged during engine out operations.
Autothrottle ARM Mode
The autothrottle ARM mode is normally not recommended because its function
can be confusing. The primary feature the autothrottle ARM mode provides is
minimum speed protection in the event the airplane slows to minimum maneuver
speed. Other features normally associated with the autothrottle, such as gust
protection, are not provided. The autothrottle ARM mode should not be used with
Non-Normal Checklists. Some malfunctions that affect maneuver speeds cause
the autothrottle to maintain a speed above approach speed.
And:
The autothrottles should be disconnected when the autopilot is disengaged.