I am presuming you are talking about thermodynamic efficiency here (mechanical power out divided by electric power in), not cost effectiveness. I also presume that you are talking about using a brushless motor with some kind of rotor angle feedback -- otherwise you really could not talk about substituting it for a brush DC motor.
In that case, I am hard put to come up with a scenario in which the brush motor would be more efficient than the brushless. Fundamentally, the mechanical commutation of these brush motors is quite primitive, and you will always have current in some armature windings that don't generate much useful torque due to their orientation at the time. With any decent electronic commutation scheme for a brushless motor, whether six-step or sinusoidal, phase current will be zeroed or at least greatly reduced at these times.
Other factors in favor of brushless motors: Equivalent stator windings will run cooler, and hence with lower resistance, than rotor windings because of lower thermal resistance to the outside world. Also, you don't have the mechanical friction and added electrical resistance of the brush/commutator-bar contact.
Can anybody else think of factors in favor of higher efficiency for brush motors? I can't.
Curt Wilson