During the convocation scene, the headmaster brags that over 75 percent of the school's graduates went on the "the Ivy League." In 1959, the Ivy League was only a five-year-old football conference. The concept of the Ivy League as a specific group of select academic institutions was still many years into the future at this point.
The drum cadence used by the marching band, and the style of dance displayed by the football players entering to it, are not from the 1950s. This is the only anachronism that screamed out at me during the movie, probably because I was in a marching band 10 years later, and we still would not have been playing such a cadence, nor did anyone dance that way -- especially not a football player.
When Keating is talking to his class about Shakespeare, McBeth and other poets, there is a close shot of one of the boys laughing. You can plainly see that the boys braces on his teeth are the more modern type where the bracket for the wire is glued to the outside of the tooth. In 1959, this had not been invented yet and kids wearing braces on their teeth would have had metal bands that were cemented to the whole tooth, covering it almost completely. The wire bracket would have been on the outward facing side of the band to support the wire for the straightening of the teeth. The newer version of the bracket being stuck only to the exterior side of the tooth would not have been developed for another 20 years.
When Todd receives the desk set for his birthday, and Neil and Todd go to throw the desk set off the bridge, the desk set is clearly covered in plastic shrink wrap. However, when the desk set is thrown, papers and pens still go flying everywhere.
When John Keating has the boys stand on his desk, he gathers his things to leave the room. Behind his desk (waiting to stand on it) are three boys on the left and none on the right. In the next shot of his desk, there are two boys on the left and two boys on the right.
At the beginning, after the procession arrives where the headmaster is, we can see the two first boys going to sit. Then, when the man holding the candle approaches the pew, we can see them going to sit again.
At meal time, when Mr. McAllister is talking to Mr. Keating, they are holding a tray. After exchanging them, when the camera does a close shot on Mr. McAllister, he doesn't seem to hold the tray anymore.
When the students are stepping up onto Mr. Keating's desk to get a different view of the room, Pitts goes up twice: once towards the beginning and then later when Mr. Keating tells them that they are to write a poem.
While Keating addresses the class for the first time outside their classroom, he reads out Pitts' name to ask him to read out a passage. Pitts raises his right hand, but in the next shot, Pitts is raising his left hand.
Owing to the huge popularity of the movie. There was a short movie sequel planned with Todd Anderson following the footsteps of his mentor. But was canceled due to the unavailability of the original cast.
When Neil is reading from the introduction to the poetry book, and Mr. Keating is drawing the diagram on the board, Cameron starts to copy the diagram down in his notebook. Cameron draws and shades in the boxes on his notes before Mr. Keating ever draws them on the board.
The bagpiper in the first scene takes his pipes out of the case one moment and is seen playing "Scotland The Brave" during the ceremony apparently the next, without any indication any time had gone by. In reality it takes several minutes to tune a full set of pipes to the tonal qualities expected at a graduation (or similar).
The Thoreau quote read at the beginning of each meeting is incorrect. In fact, if you look very closely, in the scene where Neil opens the book that Mr. Keating had placed on his desk, you can see three dots after some lines, indicating that the sentences had been taken out of context. However, it is indeed "I went to the woods because I wished to live deliberately", and not, as quoted in the film, "wanted".
The school's name is Welton throughout the movie. Various characters call the school "Hell-ton" including the cafeteria's offering of "Hell-ton Hash". It's not a mistake about the name of the school, just boys making fun of their school.
The goof item below may give away important plot points.
When the boys are standing up on their desks, making a tribute to Keating, a glimpse of Charlie/Nuwanda as being one of the boys standing up can be seen. However, as pointed out a few scenes earlier, his character was expelled from the school so there isn't any reason why he'd be there at the class. This goof was seen in theaters in 1989 and now can only be found in the earlier video versions and sometimes on cable presentations, but it's no longer present on DVD and Blu-rays.