The United State Postal Box pictured in the opening scenes has the new eagle symbol showing only the stylized head of the eagle and not the the eagle with the raised wings such as would be in the early 1990s when the film was based.
In one shot of Micky Ward's apartment, a bag of teriyaki Jack Links beef jerky can be seen on an end table. The design on this particular bag wasn't unveiled until Jack Links re-branded their company in 2005.
When Micky Ward takes Charlene to see the movie, in the background you can see a Citizens Bank. The logo, which is that of the Royal Bank of Scotland, wasn't adopted by Citizens until approximately 2005 or later, though the movie takes place in the 90s.
When the family is all gathered in the living room, there is a Rock 'Em Sock 'Em Robots sitting on the end table next to Micky Ward. Then it's gone when they cut back to him. Then it's back, then gone again.
During the Shea Neary/Micky Ward fight, the boxers were weighed in at "10 stones 6 pounds" each. 10 stones equal 140 pounds. That would make them both Welterweights. In reality, they fought at Junior Welterweight (limit of 140 pounds). At the actual weigh in 2000, Neary weighed in at 139 pounds and Ward at 140 pounds, the limit for Jr. Welterweights.
Both Micky Ward and Dicky Eklund appear together in court before Dickie went to jail for multiple serious charges. In reality, Micky was arrested while interfering the arrest of Dickie, who committed a relatively minor offense.
In the opening scene where Dicky Eklund and Micky Ward are walking through the streets while getting filmed, one of the onlookers has a tattoo with the 'new' NE Patriots logo. While probably an error, this logo was first used in 1993 the year this scene takes place in.
The song "Strip My Mind" by Red Hot Chili Peppers from 2006 is played during the montage of Micky's training and Dickie's prison stay. It's only incidental music, and not heard by the characters in the story.