Some of the keyboards and mice on various desks weren't in production when the film takes place, and some didn't even enter production for nearly a decade after the events of the film. For example, the pair on Ben Bradlee's desk were included with Dell desktops released well after 2010.
When Mike Rezendes enters the Globe after getting out of the cab, a Star Market sign is visible behind him. The supermarket near the Globe headquarters was a Shaw's in 2001. It became a Star Market in 2011.
When Sacha Pfeiffer is conducting an interview with a victim in the coffee shop, she takes notes on her steno pad. At one point only a few lines are written, the next shot shows the pad full of notes, and a third shot shows a few lines written.
When Rezendes and Garabedian chat outside the courthouse while Garabedian eats lunch, a tall, blonde woman in a gray business suit walks past them away from the camera at the beginning of the scene, and again at the end of it.
When Mike Rezendes and Mitchell Garabedian talk outside of the courthouse, and Michell tells Mike about the 14-page document, Mike has his shoulder bag. The bag disappears in the next scene, when Mike rushes back to the Globe office.
When Sacha Pfeiffer is interviewing Joe Crowley in the park as they walk down an incline, a man and a young girl walk right to left in the background behind a cast iron fence. When the camera angle changes, the man and girl are gone.
At 12:00, Walter and Mike are talking to Ben in Bradlee's office. Bradlee pulls his glasses out of his shirt pocket and tosses them to his right on the desk as he talks on the phone. The camera cuts away for just a few seconds, but when it's back on Bradlee, the glasses are back in his shirt pocket.
During the cab ride, after Michael Rezendes pays for copies of the sealed documents, he tells the cab driver to take Dorchester Avenue and stay off I-93. The cab is never on Dorchester Ave, and hits I-93 south.
In the next to last scene, the two reporters show up at the Globe and park next to each other. Both of their cars have Massachusetts plates with 7 digits. Massachusetts never has more than six digits on their license plates.
Ben Bradlee jokingly claims to be the Archbishop of Canterbury when Mike is spooked and asks who is at his door. The Archbishop of Canterbury is the head of the Church of England and would likely have little interest in protecting the reputation of the Catholic Church.
Some characters use the term "9/11" to describe the terrorist attacks of September 11, 2001, which were very recent. However, 9/11 was used almost immediately afterward. On September 13, 2001, the New York Times announced The New York Times 9/11 Neediest Fund, a "special campaign to raise money for the victims of the attack on the World Trade Center."
At the court house, Michael Rezendes gives all his money ($83) to the clerk so he would make copies of the documents for him. In the next scene, he rushes out of the building, gets into a cab, rides to the Globe building, and runs inside. Unless an ATM was inside the court house, or they stopped at an ATM on the way to the Globe, he wouldn't have had any money to pay the taxi fare.