When the family of J Paul Getty II goes shopping in San Francisco in what is supposedly 1964 according to the captions leading up to the scene, there is a maroon first or second generation Mercury Cougar in the parking lot of the store. The earliest Mercury Cougars were released for model year 1967.
Also (this is not a goof), one wonders how a family that is supposedly tight financially could have an eight year old Chevrolet Wagon in such straight and clean condition with no faded paint, dents or missing parts visible.
When Gail is showing Fletcher Paul's room, a poster for the movie "Lo straniero senza nome" can be seen on a wall. It's the Italian title of "High Plains Drifter" by Clint Eastwood. The movie was released in August 1973, while Paul Getty was kidnapped in July 1973.
As well as the 1977 Apple II computer previously pointed out, there is also a 1977 Commodore Pet visible in the background of JPG's library, again set in 1973. Even the 6502 processor chip that they both use wasn't produced until 1975.
In the early scene where the family has breakfast there is a glass bottle of juice visible, which change amount of juice back and forth between each different camera angle, less juice, full juice, then less again, then back to full etc.
When J. Paul Getty is being interviewed by Playboy magazine, Getty is sitting opposite the interviewer and is fondling a dog that's sitting to Getty's right. When the camera switches back to the interviewer while they're still talking, we see the dog in the foreground moving around in front of Getty. When the camera switches back to Getty, without any break in the conversation, the dog magically appears once again at Getty's right-hand side.
Just after Paul is recovered, the car containing him pulls up to a tollgate that is swarming with paparazzi. From behind, you see the car pull in, stop, and the tollbooth worker takes a document from Chace, who is driving. The view switches to the front, and the car is seen pulling into the tollbooth again, and re-handing the document to the tollbooth worker.
The end of the film depicts J. Paul Getty dying at the same time Fletcher Chace and Gail Harris rescue J. Paul Getty III. In reality, the elder Getty died in 1976, three years after the kidnapping of his grandson.
The villa model J.Paul Getty is showing Fletecher near the end of the movie is the Getty Villa in Malibu. This villa was not based on the Villa Adriana at Tivoli, that is shown being visited by J.Paul and his grandson at the beginning and mentioned again by J.Paul to Fletcher. It was in fact a reconstruction of the Villa of the Papyri of Herculaneum. This smaller mansion was destroyed in the same eruption that destroyed Pompeii in 79 CE, and was not a imperial residence, but probably the property of a well to-do Roman patrician.
11 minutes into the movie, in a scene that occurs during a period identified as 1964 and showing per-Christmas preparations, we see a snow storm in San Francisco, CA. The only snow in San Francisco in the 1960s was on January 21, 1962, almost two years earlier. but the snow fell in the Sunset and Westlake Districts, There was no snow in the financial district in any year of the sixties.
The Carabinieri attacking the kidnapper's cove are seen handling a number of different firearms. One of the them is a Beretta AR-70/90 assault rifle. This is an assault rifle introduced in the 80s and used only by the Italian army proper, not the Carabinieri.
When the boy is shooting the rock in his slingshot the car is broadside to him, even a little past when he releases the shot. The rock still manages to strike the windscreen, which is impossible from that angle.
The goof item below may give away important plot points.
The car outside Gail's apartment in Rome had a red star and circle painted on it's roof. When Mr. Chace is tapped by the car that tells him to follow, he pulls out and the red paint is no longer seen, the roof is again white.