The mushrooms that contain psilocybin and psilocin will not cause liver damage.
There are other mushrooms that will cause liver damage if eaten. They contain toxins called amatoxins. The mushrooms which contain these toxins are mostly from the genera Amanita, Galerina, Lepiota, and Conocybe. Amatoxin poisoning is responsible for over 95% mushroom poisoning fatalities. Death is usually in 7 to 10 days due to liver failure.
The cases related to psilocybin containing mushrooms causing liver damage are cases where somebody misidentified one of the deadly mushrooms mentioned above for one of the psilocybin containing species. The most common mushroom for this is Galerina marginalis. Sometimes the deadly Galerinas are mixed in with active Psilocybe or Panaeolus species. There are photos documenting the two growing side by side taken by Mushroom John (mjshroomer).
Other species such as Gymnopolis contain Psilocin but also contain other substances with the potential to do liver damage.Some of the other lesser species which contain psilo(cybin)cin either come from genera which contain deadly relatives or are so low in concentration the amount needed to achieve efect could be also supplying a large dose of a toxin. In short the three main active genera containing Psilo(cybin)cin, Panneolus, Copelandia and Psilocybe do not exhibit any liver toxic properties.
The Amanita muscaria comes from a genus containing some very poisonous fungi known to cause liver failure. It has also been shown there is a marked difference in effect in these mushrooms from collection area to area and person to person. Liver toxicity is a possibility with some Amanita species.