Sue's hairstyle and hair color change throughout the movie because some scenes were re-shoots in which she wore a wig.
Reed's facial hair constantly appears and disappears throughout the film because certain scenes were re-shoots.
When Reed and Sue are talking in the library, Sue takes one of her earbuds out, and it remains out while she talks to Reed about him wanting to be famous, but it suddenly returns to her ear when the camera cuts back to her.
(at around 57 mins) The Human Torch is shown blowing up an MQ-9 Reaper drone for target practice. These are active, front-line drone aircraft which are crucial to current U.S. military operations, and at $17 million each, it would be absurd to waste them for target practice. The U.S. military actually uses retired fighter aircraft converted into drones for aerial target practice, such as the QF-4 Phantom II, or the QF-16 Fighting Falcon.
(at around 4 mins) When Reed is in the junkyard, he claims to be looking for a "power converter", but leaves with an electrical transformer, which isn't the same thing.
(at around 57 mins) The MQ-9 Reaper drone used as target practice is showing flying at extremely high speeds and being highly nimble and maneuverable. It is actually a slow, propeller-pushed aircraft with a cruising speed of only 194mph (313kph).
(at around 1h 5 mins) The U.S. Army Special Forces troops who attempt to capture Reed Richards are supposed to be rappelling from a hovering helicopter, which is conveniently off-screen. The effect of being directly below a hovering helicopter is shown as a mild breeze. The downdraft of a helicopter is far more intense.
(at around 1h 2 mins) When Reed purchases what he says is a valve, the object is actually an old style fuel pump for a carbureted engine.
Young Reed has blue eyes while older Reed has brown eyes. Blue-eyed children sometimes grow up to have brown eyes.
(at around 13 mins) Flipped shot. When Reed is looking through the Baxter complex library, several shots of him looking through the books show his scars on the wrong side of his face.
The U.S. Army soldiers at Area 57 are wearing unit patches of the 8th Infantry Division. This unit was deactivated in 1992.
(at around 41 mins) No explanation why the flagpole had such an immediate and dramatic effect, but neither the piton nor the repeated arrival & departure of the shuttle seemed to disturb the ground at all.
Where does Debs suddenly come from when she appears on the bridge?.
(at around 1h 3 mins) Ben is shown as a 10-year old 5th grade student in 2007, but in 2015 his military file lists him as born in 1986 (which is, in fact, his actor Jamie Bell's birth year). If that were true, it would mean that Ben has failed the 5th grade 10 times and has not hit puberty by the time he's 20 in 2007, which seems highly unlikely.
(at around 1h 3 mins) Reed reads government information on Ben Grimm aka "The Thing". The information states that his eye colour is black, when his eyes are clearly blue.
(at around 34 mins) When he is offered an alcoholic drink, Reed says, "Ethanol kills brain cells," which isn't true.
(at around 34 mins) When Dr Allen is talking about bringing in NASA, Victor makes a remark about "waterboarding in the 4th dimension". Any good scientist would know that the 4th dimension is time and isn't a physical place.
(at around 13 mins) Reed shows Sue "20,000 Leagues Under the Sea", commenting that Nemo invents a submarine that goes deeper than anything in history. This suggests that Reed doesn't know that the 20,000 leagues refers to the distance traveled while under the sea, not the depth. The rest of the movie suggests he's smarter than that.