In the talent show scene during Sparkle Motion's dance (1:12:50), Mrs. Farmer is seen backstage standing next to a theatrical stage light. That stage light, an "ETC Source Four Par", wasn't invented until 1995, yet the film is set in the year 1988.
During the Life Line activity in Mrs. Farmer's class, Cherita answers her question card by placing an "X" on the chalkboard. When Donnie is called next the "X" that Cherita placed on the board is barely visible, like it had been half-erased.
During life line exercise activity when Ms. Farmer pulls the board into class, there is a black duster on the board just above the rectangular shape. When she starts distributing life line exercise cards, the duster shifts its position to right. When Cherita comes to the board, it's back in its original position.
When Donnie is talking about time travel with his science teacher for the first time, the slinky around his neck changes positions from behind the collar of his shirt to around his neck when the camera switches between frontal and rear angles.
When Donnie's science teacher begins talking about Roberta Sparrow's book, Donnie puts the Slinky around his neck. The camera cuts to Donnie taking the book and cuts directly back to a rear angle of Donnie to show the Slinky still taut around his neck, which it would not be if he had released his grip on one end of it.
In the final scene between Donnie and his therapist, Dr. Thurman, Dr. Thurman appears to have had a hair-cut, as her hair was shown earlier in the film as around her shoulders in length. However, in a scene that follows soon after, when Dr. Thurman calls the Darko house, her hair is seen to be at its original length.
Elizabeth receives her admission letter to Harvard in October, before Halloween. (The party she and Donnie have is a Halloween party to celebrate her admission.) However, Harvard belongs to the universities that use the universal application and notification dates: applications have to be received by Januay 1 and letters of acceptance (or rejection) are mailed on April 15. To be coherent with Harvard's system, the film would have to take place in April, not October.
The name of the high school is "Middlesex," which would obviously be abbreviated "MHS" for Middlesex High School. Immediately before Donnie's time travel conversation with his teacher, we see the top of the entrance of the school. There is a stone cross there with the letters "IHS". However, "IHS" is catholic acronym from Greek name of Jesus (Ihsous) and is commonly placed on the crosses. This letters aren't supposed to be acronym for the name of the school.
The movie is set in 1988, but license plates, book editions, news photographs and products from 1989-1999 appear throughout. These anachronisms fit in with the movie's themes of time travel and altered realities. (Chapter 4 even mentions how things from the future, such as Mayans having arrow heads, are symbols of future being brought to the past - these anachronisms are on purpose.)
When the students all write "They made me do it" on the board, Sam Bylen is followed by Donnie Darko. Cherita Chen wrote between those two, but the original cut skips her moment. It is restored in the director's cut.
When Donnie finds the 1911-style pistol in the closet the hammer is not cocked and the safety is on. It is impossible to engage the safety on a 1911-style unless the hammer is cocked. And there is a screw or pin visible in the rear left side of the slide which does not exist in a real 1911.