At some point during the interrogation/torture of Steven Younger, virtually all of his fingers on at least one hand get amputated. A couple of scenes later, all of his fingers appear to be back in place, as you can clearly see him clutching the arms of his chair.
Every time 'H' open the bottle to pop his pills, the bottle is always still full.
At the very beginning, as Helen crosses the street, she adjusts her jacket because the butt of her gun is showing. The gun is located on her right side, made for a left hand shooter. In the very next scene she is seen writing with her right hand.
Although the shopping mall bomb appears to have been detonated at least 1000 feet away, the people on the roof see it and feel the blast at the same time. Since light travels much faster than sound, the blast should have arrived at least a second later.
There's only about four pounds of "nuclear material" for each bomb, but critical mass for any radioactive element is nearly three times that. Plutonium bombs overcome this with carefully shaped and timed explosives to make their material uniformly more dense. However, the manufacturing challenges are significant and no such wiring or setup is evident in the bombs.
After a torture session, H tells Brody to wait with her questioning until the adrenaline wears off in the prisoner, which she should recognize by him starting to weep. Then he tells her to wake Younger with an electric shock in case he falls asleep. If tapering off the adrenaline level is so crucial to the process, it makes no sense to shock the prisoner because that would immediately trigger another adrenaline release.
In the end the FBI agent deactivates one of the bombs with an Excel spreadsheet.
Near the beginning of the movie, there is a map with suspected terrorist cells and potential targets. The location for Cheyenne Mountain Air Force Station is incorrectly marked as near Cheyenne, Wyoming. It's actually outside Colorado Springs, Colorado.
In the old school, Brody's shoes change from pumps with pointy heels to pumps with thicker heels several times.
In Yusuf's video displaying the nuclear bombs, the radioactive material container has the word "Chernivtsi" written in Ukrainian language. Chernivtsi is a city in Ukraine, near border with Romania. Origin of bomb's radioactive material is supposed to be in Russia, making a clear geographical mistake, besides there is no nuclear power station or radioactive fuel process plant in Chernivtsi region.
When they finally discover the bomb location, a SWAT team starts running towards the bomb pointing guns at it. This makes no sense. The bomb will surely not raise its hands and surrender.
After the present has been established as Wednesday, the FBI agents see the tape in which it is stated that three bombs will go off on Friday. Agent Vincent then says "But that's 4 days from now".
When she goes in to talk to Yusuf, she says they can't prosecute someone without fingernails, what she should have said was fingerprints. Both are wrong.