CPT seems to imply it. You can reverse the system evolution by applying charge, parity and time conjugation, so the information about the past must be contained in the present state. That implies conservation of information by the evolution.
This may not be the answer you wanted, because it does not imply unitarity, but it is the only relationship between symmetry and information conservation that I can think of. Unitarity seems to be a very fundamental assumption though, and there is not much more fundamental mathematical structure you could use to argue about its necessity.