Was the Poseidon a U.S. Navy submarine?

No - there has never been a submarine with that name in the current or past Naval fleet. The only ship to bear the name Poseidon was USS Poseidon, (ARL-12), an Achelous-class repair ship in WWI.

There was a movie, USS Poseidon: Phantom Below, which had a fictitious Navy submarine called the Poseidon. However, the hull number is for the active commissioned submarine USS Georgia (SSGN 729), and there has never been an attack submarine with that hull number or designation.

Hollywood rarely uses real hull numbers of active ships to avoid any legal problems. The exceptions are those in which the production involves ships/submarines in which the storyline actually involves them. For example, The Hunt for Red October featured the USS Dallas, a real LA-class fast-attack submarine, of which an old friend of mine served aboard during filming. Usually though, the film will use a decommissioned hull number - many older films use 593, the hull number of the Thresher, which sank in the early 60's.