My idea: It will just cost energy initially, to overcome the initial resistance to change of flux by the induced current in the loop, and also to accelerate the weight of the loop. After that, it will become like a pendulum or see-saw: like potential to kinetic energy and vice versa wrt the loop as a classical mechanical object. And also potential to kinetic wrt the electrons/current.
Assuming there is already current in the coil. There is the coil current to magnet B-field alignment, which produces a force on the coil when they are not perfectly aligned. However, this force on the coil does not cause acceleration to the electrons when the coil is static mechanically, even if there is current, since it is always acted perpendicular to the motion of current, so the force only affects the coil mechanically.
Change in current or acceleration of electrons is caused only by the motion of the coil i.e. change in flux.
Now, assuming there is already existing current, a mechanical change in position of coil that is against the current to B-field alignment will cause an induced current that will add to the existing current even more. However, a change in position of coil that improves the alignment will lessen the existing current (meaning decelerate the electrons), or even eventually change the direction of current.
So we can see, everytime the coil is positioned against the direction of the alignment, there is mechanical force resisting against that direction. Now, if the coil is rotating against that resistive force and moving away from alignment, there is deceleration of the coil. That deceleration or loss of mechanical energy becomes acceleration of the electrons, since in fact it is strengthening the existing current everytime it opposes the resistance.
And when the coil is positioned against the alignment, but rotating toward the alignment, there is acceleration, increase in mechanical kinetic energy of the coil since it is going in the direction of the force. It corresponds to deceleration of electrons and decrease in current, because of the direction the coil is rotating toward alignment.
Now the case when electron deceleration stops, and current eventually changes direction, there will be a new current to B-Field alignment configuration because of the change in current direction. And the things said above still applies. If the coil initially accelerates mechanically while the current decreased, then when current direction changes, the coil will begin to decelerate, and the current will continue to strengthen in the new direction.
So apparently, energy is conserved, assuming the coil is friction-less in rotation. Energy is just converted back and forth, from mechanical energy of coil, to energy due to motion of electrons, and vice versa.