(at around 1h 15 mins) In the flashback scene when Sir Edmund tells about Arthur and his knights working together with the Transformers, the environment where the Round Table is placed is shown as a Gothic style cathedral ruin. In reality Gothic style didn't appear until the 12th century, so cc. 700 years later. The clothing also show similar anachronisms.
(at around 33 mins) Daytrader brings Starscream's head to the junkyard, and Megatron is later seen holding it when he and his Decepticons arrive. However, the head here is fully intact, when in Transformers: Dark of the Moon, Starscream lost his eye and his head was blown apart by Sam.
(at around 22 mins) Optimus arrives at Cybertron and is distressed, asking "what happened to [Cybertron]?" However the fate of the planet in Transformers: Dark of the Moon would have been clearly visible to Prime back on Earth.
(at around 1h 12 mins) A statue of Optimus from ancient history is shown in Sir Edmund's office. The design used for Optimus here is clearly his knight armor that he wears in this film, despite the fact that he did not look like this until the fourth installment.
(at around 48 mins) When the Autobots are leaving the junkyard before the Decepticons arrive we see Squeaks with both his original arms. He had one shot off back in Chicago and Izabella attaches a Decepticon arm to him.
(at around 2h) Stonehenge as shown is not as isolated in open countryside and from main roads as portrayed - the A303 among other roads runs past the monument and also the town of Amesbury is across said main road less than two miles away.
(at around 1h 40 mins) The ESA scientist says that "the planet hid its approach behind our star", however the shot of Cybertron immediately after shows it is moving from outer space towards Earth and the Sun.
(at around 1h 22 mins) Hopkins is shown running down St Helens Passage in Oxford to get to the library. Not only does that mean he's running away from the library, he's also going down a dead-end that leads to only one thing - a pub (the sign for which is clearly visible).
(at around 1h 35 mins) While the submersibles are following the submarine into the alien space ship the Navy Seal says "we are about to lose radio contact".... radio waves cannot pass though water therefore radios are not used for under water communication.
(at around 2h 5 mins) Towards the end of the movie Colonel Lennox states that the Osprey's flight ceiling is 12,000 ft and the target where the battle will be taking place is at 21,000 ft, "nearly double" and the air would be too thin for the Osprey's rotors to "bite". This is incorrect. The Osprey's operational flight ceiling is in fact 25,000 ft, well within the operational area.
(at around 28 mins) When we first see the Samurai Autobot "Drift" in the junkyard he is balancing on his sword on the roof of a car causing no damage apart from a few scrapes. The sword alone would have been heavy enough to at least dent the roof, adding the weight of Drift would have caused him to skewer the car like a kebab.
(at around 1h 28 mins) At one point, during the car chase through the streets of London, one of the Police vehicles is flipped and the branding on the roof of the vehicle can be seems as 15 LTS. This code is for Police helicopters to easily identify the specific unit. The first two digits are force codes. The Metropolitan Police cover London and their code is 01, not 15.
Throughout the movie military characters refer to alien attack aircraft as "bogeys". This is common in Hollywood movies but is wrong: a "bogey" is an unidentified aircraft, i.e. usually an aircraft caught on radar that has yet been identified as friend or foe. An enemy aircraft is called a "bandit".
(at around 45 mins) When Daytrader brings a new voice box for Bumblebee, he lies on his back like a patient at the dentist while Cade and Izabella fix it for him. In several shots, Bumblebee's head is clearly a prop as the rest of his body has disappeared behind Cade and Izabella.
(at around 1h 27 mins) At one point during the car chase at Buckingham Palace, bystanders are shown watching and/or filming the scene. Likewise, the barriers blocking the streets and sidewalks for filming are visible.
The goof items below may give away important plot points.
(at around 1h 50 mins) The brainwashed Optimus regains control of himself once he hears Bumblebee speaking with his true voice. Prime claims that he hasn't heard Bee's voice since Cybertron. However, Bumblebee used his true voice at the end of the first movie, while still on Earth.
(at around 1h 55 mins) When Bumblebee and Optimus are fighting on top of the alien ship, Cade's shirt is torn around the neckline and the arm. After Cade saves Optimus from the Knights, his shirt is no longer torn.
(at around 1h 29 mins) When Bumblebee shoots Barricade in London, driving past a side road, you can see a normal police car next to a regular car waiting, but in the close up of the Decepticon crashing/blowing up on to that street the cars have changed.
(at around 1h 40 mins) When Cybertron is approaching Earth, a scientist from the European Space Agency states that the International Space Station will be hit by the planet and that it has no escape pod. This is inaccurate as the International Space Station has a one or two Russian made Soyuz spacecraft docked during an expedition depending on how many astronauts are on board. So in reality, the astronauts would have been able to escape and avoid certain death from collision between Cybertron and the International Space Station.
(at around 1h 40 mins) When Cybertron scrapes the surface of the Moon, the landing site of the Apollo 11 Lunar Module is shown with the American flag still standing. This is wrong as the flag was blown over when Apollo 11 lifted off from the surface due to it being planted too close to the Lunar Module. Future flags were planted further away during the other Apollo missions to prevent this from occurring again.
(at around 51 mins) The talisman attaches itself to Cade's right arm at the bottom of the rolled up sleeve. We see that arm several times over the next several minutes and the talisman is not there. The talisman is later shown to be moving around Cade's body, and could just be out of sight at the mentioned times.