The tail rotor is there for two reasons. One is to counter the torque on the main rotor, the second is for yaw control.
A counter rotating mass could do the job of countering the torque, but at a cost: weight. If the mass turned at the same speed as the rotor and the mass was a ring about 4 feet radius with a 20 foot radius rotor (pulled numbers out of air), then the counter rotating mass needs to weigh about 17 times more than the rotors. If the mass turned twice as fast, it would weigh 1/2 that. A transmission is also needed, the faster you spin the mass the beefier the transmission will have to be.
That would counter most of the torque, but it wouldn't automatically adjust to every condition of flight. That's okay because you still need yaw control.
Conventional yaw control wouldn't rob as much power as it does now, but the weight of all the components won't change.
Yaw control in the counter rotating mass is possible if you have movable masses inside that could be moved radially using the yaw control pedals. I doubt if this would be easy to control...there would probably be a noticeable lag between input and response.
Of course, the engines must spin that counter rotating mass, so efficiency may actually drop.
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In short, it is possible. My new cordless screwdriver uses this technology. I doubt if any overall increase in efficiency will be realized.