When Tom and Dickie are in the boat on the sea of San Remo, in the background is visible Casinò d'Anzio situated in town of Anzio, some 60 km south of Rome. It's the same building of the previous jazz session scene.
When Marge arrives in Venice at the Santa Lucia railways station you can spot in the background on the other side of the canal one of the very few modern buildings of Venice. It is obvious that this building was built much later than 1958 when the story is supposed to take place.
When Damon talks to Paltrow after he returned home from San Remo, we can see the cover of a Miles Davis LP in the background. It is the cover sleeve of "Tutu", released in 1986. The movie is set in 1958, however.
The ship Tom takes to England is the Queen Mary, shown passing the Manhattan skyline. But the lifeboats are covered with blue canvas tops, not installed until the ship became a hotel and convention center in Long Beach CA.
When Ripley and Dickie are playing chess, Dickie moves a pawn. A
bit later, we see a close-up of the chess-board, where Dickie makes the same move again. In addition to Dickie making the same move twice, other pieces change position in different takes in the chessboard scene.
When Freddie and Dickie are talking on the boat, Freddie has a drink in his left hand, and stirs it nervously with his right. From the reverse angle, the drink is in his right hand and there is no stirring.
In the Hot Jazz Vesuvio club when Dickie invites Tom onto the stage, Dickie moves to one side. He then licks his lips and moves his sax up to his mouth to play but in the next shot the sax is away from his mouth.
When Tom Ripley is trying to learn to identify jazz musicians from recordings, he's listening to Charlie Parker and Dizzy Gillespie's "Ko Ko", and he "recognizes" Parker's playing during Gillespie's break.
When Ripley and Peter take the ship to Greece, the sun sets to the port side of the ship, indicating that the ship is traveling roughly northwards instead of eastwards as it should be if it's sailing from Italy to Greece.
When Tom Ripley is at the theatre in Rome, there is a shot that slowly zooms into Toms face. At the close up you can see that there are no lenses in his glasses, this was probably done to avoid the reflections from the film lights.
The 'glasses' Ripley wears throughout the film often do not have lenses in, revealing them as a prop. This would be explainable if they were being used as a disguise, but Ripley always puts them on before reading or using his binoculars, meaning he needs them.