I'v never seen any conclusive research that they do. In the book "Saying yes, in defense of drug use", it was stated that about 0.08% of LSD users experience profound mental changes, and even then, it's highly unrealistic to assume that it was due to the drugs themselves, and that these people didn't have the illness to begin with. From what I gather, many mental illnesses effects are triggered by profound events in a person's life, and psychedelics are no walk in the park.
Also, at least in the case of LSD, the chromosome breaks that occur in the brain are only temporary, so it would be unrealistic to say that it would actually cause an illness. Perhaps trigger, but not cause.