Is the inverse of a causal system also causal?
It's not sufficient to only consider causality, you also need to check whether the inverse system is stable, otherwise it can't be implemented. If $G(z)$ has zeros on the unit circle, it cannot be inverted. If $G(z)$ has no zeros on the unit circle, but if there are zeros outside the unit circle, then there is no causal and stable inverse, because the zeros of $G(z)$ become the poles of the inverse system. There is, however, a stable inverse but it is not causal. If $G(z)$ has all its zeros (and poles) inside the unit circle, then there exists a causal and stable inverse (with all its poles and zeros inside the unit circle). Such a system is called strictly minimum-phase, and its inverse is also a strictly minimum-phase system.