When Vito with his family leaves Corleone to go back in the States, he starts his trip taking a train in the Corleone station. Corleone never had a train station, the nearest (34 km) useful train station is in Palermo, and is not looking like the little village station we see on screen.
When young Vito arrives in New York and the Statue of Liberty is shown, the patina on the statue is clearly bright green. The patina on Lady Liberty would not have been completely developed in 1901. At the time, the patina was covering most of the statue, starting with the torch, arm, head and torso, but not the entire length of the statue's dress as shown in the film. It should have been covered in a patchy patina from the waste down, with a primarily brown color.
The ship on which young Vito Corleone arrives in the United States during 1901 is The Moshulu. But the Moshulu was not launched until 1904 and did not travel to the U.S. until 1914. The original name of the ship was The Kurt. It was renamed The Dreadnaught in 1917 and shortly after re-named The Moshulu.
The Chairman of the Senate Investigating Committee asks Michael, "Is it true that in the year 1950, you devised the murder of the heads of the so-called Five Families in New York?" But at the party in the opening scenes, which takes place in 1958, Kay reminded Michael that, seven years previously, he had promised her that "within five years, the Corleone family will be completely legitimate." If that conversation took place in 1951, then the murders the Chairman referred to, which came afterwards, could not possibly have happened in 1950. In fact, since Michael refers to his son (who was neither born nor even conceived in 1951) being three years old shortly before the murders, it is probable that the murders did not occur before 1955.
In the Havana scene (pre-1959) where Hyman Roth is being examined in his hotel room by a physician, one can hear distinctly the sound of Velcro as the blood pressure cuff is being removed from Roth's arm. Velcro was not commercially available until the 1960s.
When Tom Hagen flies into the Nevada brothel to "rescue" Senator Geary (a common method to arrive at them at their remote parts of the state) he arrives in N3254S, it is a 1964 Cessna 182G. The time in the film is before the 1959 Cuban Revolution and a straight tail 182 would have been in production at the time with the slant tail not seen until 1964.
The aquarium at the Tahoe estate is of a type (without metal frames) that did not exist in the late 1950s - early 1960s. New silicone adhesives made frame-less aquaria possible only a few years before the film was shot.
Michael comes home to Lake Tahoe, after Christmas, his car passes through the gate, and the gate is closed by one of his security team. On the gate is an "ADT" security monitoring sign, which was introduced in 1974.
In the street scene outside the Hotel Washington after the Senate hearing, the rear of a Greyhound motor coach is visible. The vehicle in question is a GM Coach PD-4106, which did not enter production until 1961, two years later after the hearing. Further the livery (the color scheme) was not the original one applied at the factory; it did not begin to appear until 1964, five years after the hearing.
When Michael speaks with Connie at the communion party in Nevada in 1958, he reprimands her that her first-born son was picked up on a misdemeanor. Connie's wedding was in the summer of 1945. Even if she became pregnant on her wedding night, Connie's son would not have been born until 1946, making him 12 when Michael and Connie have their conversation in Nevada.
After the attempted assassination on Michael, Tom and Rocco are overlooking the discovery of the dead bodies in the water. When Rocco turns away from Tom to ask where Michael is, his lips do not move while the soundtrack says "Where's Mike?".
When Michael reads his prepared statement to the Senate committee he states that he received the Navy Cross for his wartime service; however, in The Godfather (1972) he wears the ribbon for the Silver Star instead.
The Chairman of the Senate Investigating Committee points out 1947 as the year Michael Corleone kills Virgil Sollozzo and Captain McCloskey; but in The Godfather (1972) it was established as happening in 1946.
Shortly after the assassination attempt on Michael in his bedroom, he meets alone in a room with Tom Hagen. They sit at an empty table. After talking for a few minutes, Michael offers Tom a glass of Courvoisier, from a bottle which has randomly materialized on the table.
During Roth's birthday party, the pattern on his shirt changes. Due to weather difficulties, the two minute scene took over a week to shoot and the original shirt was lost at some point. The production designer attempted to recreate it by drawing an approximation of the pattern onto a plain shirt, but it didn't quite match.
During the shooting into the Corleones' bedroom, the mirror on the dresser shows Kay sliding off her side of the bed as soon as the gunshots pierce the window and at this point her pillow is vertical. But as Michael crawls to Kay's side of the bed, she's back on top, waiting for him to pull her down, and her pillow is now laying flat.
When Michael returns from Las Vegas after Christmas, he is carrying a briefcase in one hand, and nothing in the other hand. As he walks into the house, he's holding the briefcase in one hand and a lit cigarette in the other.
When Michael and Kay are in their bedroom after the party, Michael is looking out the window, and below him is a mirror that shows Kay's reflection. When Michael ducks to avoid the wave of bullets that comes through the window, Kay's reflection disappears.
When young Vito is quarantined, we see him looking out the window at the Statue of Liberty. From the outside shot he appears to be standing at the right-most window pane, but when we see the scene from inside, he's at the other end of the window (i.e., what would have been the left side as seen from the outside).
When Michael returns to the snowy Lake Tahoe estate, we see him looking at a toy red car and then walking past a window. In the next shot, when he enters the building, he suddenly has a lit cigarette in his mouth.
At the New Year's Eve party in the Cuban ballroom, there is in aerial shot that clearly shows Michael with his hands around Fredo's face amidst a very crowded room to deliver "you broke my heart". However this had not taken place yet because Michael is eating at the end of the table with Fredo and Senator Geary *before* the midnight celebration. "You broke my heart" comes after the stroke of midnight. Also, the room was far less crowded as they were eating when a unit of soldiers march thru the middle of the ballroom.
At the end of the movie in the flashback, they talk about the Japanese attack on Pearl Harbor happening "on Pop's birthday". The Pearl Harbor attacks were on December 7, but according to his tombstone, in The Godfather (1972), Vito's date of birth is April 29.
When young Vito is quarantined, we are led to believe that he is still on Ellis Island. As he looks out the window, the shot from outside shows the reflection of the Statue of Liberty. By that reflection (the front of the Statue) Vito would have to be on Governors Island.
During the final scene flashback, they are talking about the attack on Pearl Harbor happening on "Pop's birthday". It is also revealed that this led Michael to enlist in the army "this morning". Since the attack happened in Pearl Harbor at 8AM Hawaiian time, news of the attack couldn't have reached New York before 1PM.
As Michael and his men head to Florida, there is an external shot of the front of the train they are traveling on. It is clearly a Union Pacific Railroad train with the bright yellow orange diesel engine with the UP emblem, blue wings with a red and blue shield. Their coverage is entirely west of the Mississippi River.
Vito's surname is changed at Ellis Island in the movie. In reality, this almost never happened unless requested by the immigrant. Each immigrant had to have paperwork specifically saying their first and last name, and if something did not match, they were sent back to their home country to retrieve the correct paperwork. Last names of immigrants changed on a whim (like in the movie) are a commonly believed (but untrue) myth.
In Michael's office during Anthony's communion party, it appears the Senator sets down his bottle of pills on the table after he takes them and never picks it up. However, although very brief, immediately after setting down the drink used to take the pills, you can see the Senator motion for his pills. After returning from a shot of Michael, the Senator is then seen closing his right pocket, presumably with the pills inside.)
In his later scenes, Don Fanucci has slash marks on his neck/throat area that weren't there earlier. (This is because of a deleted scene in which Fanucci is jumped on the streets by a group of young thugs, who slash him. The scene doesn't appear in the regular cut of the film, but does appear in the edit of The Godfather: A Novel for Television (1977), and as a deleted scene in the DVD set.)
When they are passing around the solid gold telephone, most of the people show how heavy it is. And yet, when Michael Corleone, and Hyman Roth handle and pass the telephone, it appears light as a feather and no indication is given as to how truly heavy a solid gold telephone is. Thus destroying the illusion that it is real.
In the opening scene when Vito's mother touches Paolo's body, his hand visibly moves. His fingers curl up and that is something a dead body just can't do. He is also seen obviously breathing in the subsequent moments.
The goof items below may give away important plot points.
After Vito Corleone kills Don Fanucci and is sitting on the steps with his family, Sonny can be seen yawning; however, when we cut to a shot of Vito holding Michael, Sonny's body is in a different position and his facial expression has clearly changed. Also, when we cut to the pan version after Vito has told Michael that he loves him, Sonny has the end of the flag out of his mouth and is sitting with his hands on his lap.
As Fanucci mounts the stairs, prior to being assassinated by Vito, the towel wrapped around Vito's gun varies between a cone-like shape and a flat-ended cylinder. In the close-up as Fanucci reaches for his keys, it's a cone; as he unlocks the door and tinkers with the lightbulb, it's a cylinder. The appearance when the gun is fired is different again.
When Michael is waiting to see Fredo (during the mourning gathering following their mother's death), Michael is wearing a watch. Then when he goes in to see Fredo (and give a nodding approval for his assassination) the watch is clearly absent.
The Lake Tahoe estate is in Nevada. The small boat that Fredo was killed in had a CF registration number indicating it is in California. The Lake Tahoe scenes were shot at the Kaiser estate that's located on the west shore of Lake Tahoe.
When Vito shoots Fanucci in the chest there is a bullet hole in his vest, however when Fanucci opens his own vest to look, there is no hole or any blood on the shirt underneath. After he is shot for the 2nd time (in the head), his shirt is all of a sudden covered in blood (supposedly from the 1st shot).
When Michael's henchman is strangling Johnny Ola, the hanger the henchman uses for the garroting slips from his left hand. Dominic Chianese (who plays Johnny Ola) has to use his own hand to keep the hanger gripped around his own neck.