Suppose you have a space elevator and you use it to lift weight to orbit. Obviously this is pulling down on the counterweight. So doesn't this mean that you need to apply appropriate acceleration to the counterweight so it does not keep moving inward? If this were the case the energy still needs to be applied somewhere. Or does the rotation of the Earth provide that energy (Earth's rotation is decreased slightly)?
The counterweight orbits further out than the geosynchronous orbit altitude at the rotational rate that's geosynchronous. That means that the counter weight constantly wants to drift further away from earth, and requires being accelerated down towards earth by the tether.
When you attach an object to the tether and then ascend the tether the tension in the tether above the object remains as it would without the object, and the tension of tether below the object is reduced by $F=m\,(a+g)$ Where a is usually small and g is a function of height. The energy to advance the object upward with this force must be supplied by the object pulling itself along tether. The force of the object pulling itself along the tether means that the tension in the tenther while there's no object ascending must be greater than this reduction (assuming a flexible tether than cannot sustain compression forces)
The energy to drive the object upward may be delivered along the tether via electricity, or may be solar powered, or it could even use chemical energy. Any power source that allows the car to pull itself along the tether could be used.
But this only covers the vertical forces. As the object ascends it'a angular momentum must increase to remain at the same rotation rate as earth. The force required to give this increase in angular momentum is perpendicular to the tether. This means for a flexible tether, that it must bend so that the tension in upper and lower portions of the tether can provide a net eastward force. This net eastward force of course also has the reaction force that pulls to the west on the earth and the counterweight, slowing down their rotation rates very slightly. The rotation rate of the counterweight is eventually restored to match that of earths by a slight angle in the tether such that the counterweight is pulled east, and the anchor is pulled west. So it turns out that all of the energy going into accelerating the object eastward does indeed come from the rotation of the earth, it's just the change in potential energy that needs to be provided by the object pulling itself along the tether.