This is because only a minimum phase system has all poles and zeros inside the unit circle on the z plane (and it is required that all poles be inside the unit circle for a causal system to be stable, just as in continuous time using the s plane, all poles need to be in the left half plane). When you invert a system, the poles become zeros and the zeros become poles, so only a causal minimum phase system has a causal stable inverse.
However, due to this possibility of equalizing a non-minimum phase system, an alternate approach that doesn’t require reducing the channel to minimum phase is using a least squared equalizer as detailed in this post: Compensating Loudspeaker frequency response in an audio signal