The story takes place in about 2005-06. Mikael and Lisbeth are using Macbook Pros with older versions of OS X, which correspond with the time. However, the Macbook Pros with black separated keys were not introduced until 2008.
Henrik stands with Mikael outside in a snow storm showing him where each family member lives and how they're each related. He points to a nice and up-kept house within walking distance of Henrik's house (the main house) and says Isabella lives there, but later Isabella's son, Martin, claims Isabella was "excommunicated to the old house on the other side of the island."
When Blomkvist is reviewing Inspector Morrell's notes from the accident, he highlights what he considers to be interesting facts. One of them is about Inspector Morrell interviewing the family, however, "interview" is misspelled "interveiws".
Stockholm has two types of subway trains in use, the old original trains and the new trains. In one clip Lisbeth is sitting in a train with the old interior, i.e. an old train. The next clip shows the train from outside (meeting another train at Slussen), but both trains are new, not old.
When Lisbeth is checking Mikael's email in the beginning of the movie, the incoming emails are all dated "2/1/06" (thus indicating either January or February depending on date setting), whereas the next scene takes place either on Christmas Eve or shortly after.
Blomkvist's MacBook Pro magically downgraded itself from Mac OS X Lion to Mac OS X Tiger during the movie. Even if he did want to downgrade the OS himself, that would be entirely impossible due to the fact that Tiger cannot run on the hardware present in the Macbook Pro displayed.
When Cecilia is looking at a family tree diagram on a wall in Mikael's cabin, the camera focuses on two paper sheets containing photos, Post-It notes and name labels about Cecilia and Harriet. When it zooms out, the photos, sticky notes and labels are arranged in a very different way.
During the chase scene; when the camera is pointing back towards Lisbeth, her face is lit by the glow of the motorcycle's instrument cluster (speedometer & odometer). But when the camera switches to her point of view; the dials are dark and not lit up.
When Lisbeth searches Hans-Erik Wennerström on Google, the search results list a Wikipedia article first that lists Hans-Erik Wennerström's birthday as "May 29, 1952." When Lisbeth clicks through to Wennerström's Wikipedia page, it lists his birthday as "12 June, 1951."
On Wennerström's Wikipedia entry, all the "blue links" are external links (denoted by an icon at the end of each link). That amount of external links in the body of an article in Wikipedia would be considered spamming and is forbidden. External links in Wikipedia (when deemed reliable and pertinent enough) normally go to a section at the very end of the article.
When Salander is searching crime records she is entering SQL queries, but the result does not match her query. The query specified last names starting with the letter "L" but the result included names that did not start with "L".