Visible crew/equipment: In the scene after Petyr's death, Deacon is standing in front of a table and the window directly above the table is shown. At that point, you can see a boom mike being yanked out of sight. I know this is supposed to be a docudrama, but this is the only place equipment is shown in the entire movie.
Suggested correction: You said it yourself, it's the boom mic of the documentary crew. They make no attempts to hide the fact that it's a documentary.
Factual error: About 20 minutes in, the group find a Welsh miners' group in the phone book. The character who is making the call speaks the first two digits of the phone number: "0, 1,..." The movie is set in 1984, when the dialling code for that district was "0639" In that year, "01" was the code for London, not Wales.
Factual error: The film makes use of a real Broadway theater, the St. James, which is on West 44th St. When they leave the stage door, the actors are next to the Edison Hotel, which is on 47th St.
Continuity mistake: In the scene where James and Ben are interrogating Jay at the pawn shop, watch Jay's shirt. It changes from green to black.
Continuity mistake: When Dex sees "his" son for the first time, the kid in close-ups is playing with the toy that dangles from above, and there's a toy football by him. In the wider angles none of the above is happening. (00:38:40)
Factual error: The "Italian" police wear obvious fake uniforms, they're US style.
Continuity mistake: During the early part of the movie, the left rear door of Doc's car is dented. For the rest of the movie, the dent is gone.
Continuity mistake: When the young girl is on the beach removing her socks, she takes three socks off.
Suggested correction: Jess is wearing yellow socks with polka dots, and yellow wellies. There are three shots in which we see Jess removing or tossing her two yellow polka dot socks, and one of her yellow wellies.
Continuity mistake: During the "Perfect Puppy" silly song, Bob places a newspaper he had been reading on the couch beside him when Larry starts singing. When Bob shows Larry the picture of the dalmatian, the newspaper has vanished. When the shot cuts back to Bob, the newspaper has returned. (00:16:55)
Continuity mistake: When the crocodile catches the man he bites his right foot off. 5 seconds later as he falls into the water his foot is back. (01:21:05)
Revealing mistake: When the newscaster was filming Clinton giving a statement, the timer on the camera stayed at 0:00. The red "Rec" button, which supposedly was on, also looked fake. (00:10:35)
Other mistake: During the driving scene in the woods with the Bronco, at least two shots are reused twice.
Factual error: In the opening heist, the gang of thieves wants to steal "The Medal of Zeus." A plate at the museum explains that in Ancient Greece, Olympic champions were awarded olive wreaths, up to the 88th Olympics in 424 BC, when a golden medal was given to the victor for the first time. This last bit about gold medals is just made up by the movie. But it also gets the timeline wrong; the 88th Olympics were in 428 BC. 424 BC is when the 89th games took place. (00:03:30)
Other mistake: In the flashback to Richard's childhood, when he looks from the door at his dad randomly shagging a party guest, the door appears open wider in the POV than it is in the other shots, where the opening barely fits his head. Besides, with the head in that position, he wouldn't see the door handle as shown. (00:02:00)
Revealing mistake: About seven minutes in, during the scene where we see Cooper helping the Nerd as a cameraman, you can tell that Cooper has been added to the scene using green-screen effects, as the actor was unavailable to shoot at the location. The lighting on Cooper doesn't quite match the lighting of the scene, and you can see a slight "blur/feathering effect" around him where the greenscreen background was removed.
Revealing mistake: When Danny is hacking Twitter through brute force, the code executed on screen is not at all a hacking tool, but simply prints jibberish - actually I googled some lines I could easily pick up without pausing, and they were from a past year finalist of the The International Obfuscated C Code Contest. Entirely harmless easter eggs, certainly not hacking or cracking material. (00:10:35)
Factual error: When the guy is talking to him on the phone and the Russian smashes it on the floor, the phone smashes and the battery comes out but the guy other end can still hear him. (01:04:00)
Continuity mistake: When Thomas puffs away from Marion, his trucks change from red to green.
Revealing mistake: When the woman is tied to the bed, she is holding the ropes, and the rope isn't tight on her right hand; she can easily escape. (00:20:10)
Continuity mistake: Jay is mopping the floor after promising on the phone to his dad that he'll bring his non-existent fiancee to the wedding. He tries to talk one of the nurses into joining him. While doing so, he stands before her leaning on his mop with his left hand on top, which switches to the right hand between shots. (00:06:40)