Moss turns up around a lot of animals, in vivariums and aquariums, on a houseplant the cat chews, on a wall the dog noses at, so it is reasonable to ask whether it is safe. For living moss the short answer is reassuring; the cautions lie elsewhere.
Living moss
The true mosses are not toxic. A cat that nibbles a terrarium, a dog that noses at a mossy log, a tortoise grazing a patch, none are poisoned by the moss itself. It carries no toxins to speak of and passes through as harmless roughage. The realistic worry is not the plant but what might be on or in it. A wild-collected clump can carry slug pellets, lawn weedkiller, fertiliser residue, parasites or the odd hitchhiking insect, so anything destined for a pet's enclosure or within reach of a curious animal should be cleaned, rinsed and ideally quarantined first, or bought from a supplier who grows it clean.
The preserved-moss caution
Preserved and decorative moss is a different matter. To stay soft and green it is treated with glycerine and often dyed, and those dyes and preservatives are not meant to be eaten. It is not classed as highly poisonous, but a pet that chews a preserved moss wall or a craft arrangement can get an upset stomach from the chemicals, and the loose fibres carry a genuine choking and blockage risk if a determined chewer swallows a mouthful. Keep preserved pieces out of reach of animals inclined to chew, and treat reindeer moss decorations the same way, since that too is preserved and dyed.
In vivariums and aquariums
Living moss is a positive for the animals it shares a tank or vivarium with. Dart frogs, geckos and the cleanup-crew invertebrates of a bioactive setup all gain from the cover and humidity it provides, and aquarium fish and shrimp graze and shelter in it. The one rule that matters here concerns what touches the water. Shrimp and other invertebrates are acutely sensitive to copper, so never introduce moss treated with any pesticide or copper-containing product, and quarantine new moss for pest hitchhikers before it goes in. Those points are covered in moss for shrimp and moss in a bioactive vivarium.
If a pet does have a nibble
A cat or dog that eats a little living moss will almost always be fine, and the usual outcome is nothing at all or a small, passing stomach upset. A pet that has chewed a good quantity of preserved or dyed moss is more worth watching, and any sign of persistent vomiting, lethargy or a suspected blockage is reason to ring the vet, taking a scrap of the material with you so they know what was eaten. None of this is cause for alarm; it is the ordinary caution you would apply to any houseplant or craft material a pet manages to get hold of.
The sensible line
Treat living moss as the harmless plant it is, clean anything wild before it goes near an animal, and keep dyed and preserved moss away from chewers. Do that and moss is one of the safer green things to have around pets, which is part of why it features so often in the enclosures we build for them. If in any doubt about a particular product, the maker or supplier can usually tell you exactly what has gone into it.