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 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.
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.
17:08-17:48 In the piano playing scene with his grandma, Oskar Schell is wearing quite dark lip color but only in the closeups. In other shots, he is wearing a lighter shade (or lesser amount of lip color).
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.
Oskar clips a particular set of names out of a phone book and is shown organizing them by geographical areas. He only clips one section per geographical area. However, names in phone books are organized alphabetically, not geographically, so there should be multiple clippings for each geographical area.
The goof items 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.
Oskar visited Abby Black on the day her husband William was leaving her. But that means Linda Schell would have visited them at an earlier date, so William might have learned then that a boy was asking about a safety deposit box key. William had already started his search for the missing key. (Although as someone else on the comment page noted, he could have hired a locksmith to open the box for him.)