Some SW fish definately bite. Some don't.
Mantis shrimp don't bite, but they do occaisionally rip open hands and shatter glass tanks with their raptorial appendages. Their diving nickname is "thumbsplitters." Triggers might bite. Eels bite. Some blennies bite. (Some are venomous, too.) Puffers might bite. Lionfish don't bite much, but they do (mostly accidently) stab people with their venomous barbs. Depends on the type of fish. If you buy aggressive fish, they are more likely to bite or sting. You are more likely to get some bad effects if you scare the fish or grab them without thinking things through.
Also, many invertebrates are venomous or poisonous, so make sure you know about corals and anenomes before you handle them much. Some "button polyps" defend themselves with lethal neurotoxin that can enter your bloodstream through cuts on your hands. At least one guy had his hands paralyzed for half a day by handling button polyps. Another person lost a family dog because he dropped an invertebrate and the dog ate it, which promptly poisoned the dog beyond all recovery.
On the other hand, clownfish, damsels, gobies, basslets, jawfish, butterflyfish, cardinalfish, dartfish, dragonets, seahorses, and most tangs seem unlikely to bite in such as way as to cause any damage, though a tang might whack someone with a tail spine.