A shot during the May Day parade sequence at the beginning of the film clearly reveals the parade to be taking place during the 50th anniversary of the October revolution (around the 1:29 mark), putting it in 1967 as opposed to 1961-63 when the story is supposed to have taken place. Therefore a person watching this parade could not have possibly defected to the USA and warned them of the Soviet missile deployment in Cuba (as is claimed in the beginning of the film).
The Kusenovs arrive in the US in a USAF C-135 marked "Military Airlift Command". The movie is set in 1962 but that organization did not come into existence until 1966. The aircraft should have been marked "Military Air Transport Service".
At Jaques Granville's party when Deveraux is setting up the meeting with his colleagues, he says, "So that's five people who can help me." The line is obviously dubbed as he clearly mouths the word "six".
While talking to the Mendozas before they leave for their picnic, in the first shot Juanita is leaning over a map with both hands on the table. In the very next shot she is standing straight with her hands in her pockets.
Andre puts the "money" in his left breast pocket, but when he goes to retrieve it at the florist, he retrieves first the drawing (which is still also in his left breast pocket), but then pulls the envelope with the money out of his right breast pocket.
In the porcelain factory, after breaking the statuette, Tamara Kusenov goes through a door to make a telephone call. The goon following her moves towards the door, now closed, to investigate it, and the shadow of a man appears briefly on the wall to the left of the door before we cut away from the shot... and it is not the shadow of the goon...
As the camera, suspended on a ceiling track, follows Rico Parra down the crowded hotel corridor in New York, one of the hanging lights can be seen to suddenly disappear upwards before it has gone out of shot (allowing the camera to continue forward).
Later in the film, as the camera pushes through the open front door into the house party, the closed door to the left of the screen can be seen to slide out of the way before it has gone out of shot (allowing the camera to continue forward).
The goof items below may give away important plot points.
When Juanita de Cordoba falls to the ground dead, her legs are bent and her dress covers her legs and feet. In the shot from the side showing Ricca standing over her body, her legs are straight out and her lower legs and feet are visible.
In the overhead shot as Juanita de Cordoba sinks down dead, her dress spreads out evenly in all directions, a brilliant metaphor for the spreading blood pool; however, once she has fully collapsed (still in the overhead shot), the positions of the wires inside the dress, which caused it to spread evenly in all directions, are clearly visible.