In one scene where Oskar and The Renter discuss 'the search', Oskar wears a turquoise hooded sweater embroidered with a small "Nike 6.0" logo (at 01:13:08; left chest). Nike did not release their "Nike 6.0" line until 2011.
The movie takes place in 2002/2003, yet at 01:10:18 in the shot of The Renter's arrow note on the bar door pointing to the door knob and again in a close-up at ~01:11:30 as Oskar peers in through a window, there's a "Recommended by Lonely Planet" sticker from 2007 on the window.
In the beginning of the film, Oskar says "There are more people alive now than have died in all of human history." There are, in fact, about ninety-eight billion dead people and seven billion people alive right now. This film is set in 2003, but the facts can't have changed much in nine years.
The continuity in editing was missed: Oskar is talking to the "Renter" in the rickety old bridge. A plane is obviously flying behind Oskar and it disappears or is out of view behind a shack for a moment. It does not reappear on the other side of the shack.
In the segment where Oskar is reading the note his father wrote, his father's voice (while reading the note as a voice over) says that Oskar has shown "unbelievable bravery", but "unwavering bravery" was what was actually written in the note.
When Oskar and his mother are recounting the people Oskar visited, after they discuss Leigh-Anne Black, both clearly refer to "Lona Black" (at 01:49:58 and 01:50:02) as she is shown in flashbacks and the subtitles show both saying Lona Black as well, but the end credits (in order of appearance) list Mona Black after Leigh-Anne Black and do not list Lona Black. Since no one named Mona Black is mentioned in the film, it is an error in the end credits.
Around 42 minutes in, Oskar remarks that "Hazelle Black lived in Hamilton Heights". However, as he is going into her house, the trash can in the garden reads "242 Eldert Street". Eldert Street is in Brooklyn.
After Oskar runs away from the Blacks in the office building he runs into the subway. He boards the number 2 train where there are multicolored seats. The 2 train did not have this pattern of seats then.
The goof item below may give away important plot points.
Oskar was obsessed with an image of a man falling (jumping) to his death who he believed to be his father. Message #6 flashback shows Oskar staring at the phone while his father asks "are you there?" over and over. The line goes dead we see the tower crumble on the TV and Oskar realizes his father has died. So he must have known his father died when the building came down, he did not jump and cannot be the falling man.