(at around 1h 8 mins) When Newt is trying to describe the physiology of the Kaiju he likens their two-brain biology to dinosaurs. However, the theory that dinosaurs had two brains was abandoned decades ago.
(at around 1h 35 mins) When Marshall is talking with Raleigh about the Mark I Jaegar's radiation affecting him, he turns around from a mirror to wipe his face with a towel. In the next shot, he his significantly farther away from the mirror than he should be, causing him to throw the towel at the mirror.
(at around 55 mins) After Mako's first drift test, Raleigh goes over to help her as she's coming out of it, there is no suit material covering her neck as she's standing. When Raleigh helps Mako to sit in the Conn-Podd her neck is covered with the suit fabric.
(at around 44 mins) When Newton is preparing for his first drift with the kaiju brain, he places the headpiece on his head. In the next shot, the collar of the headpiece has fastened around his neck without him touching it.
The scene with the Kaiju breaking through the wall in Sydney makes no sense. Given that the opera house is on the seaward side of the harbour bridge it would have had to be breaking through from the land side of the bridge. If it is already on that side it does not need to break through the wall.
(at around 1h 28 mins) As the Jaeger falls after the Hong Kong Kaiju battle, it begins to burn up/glow due to re-entry heat. In the movie, they say the Jaeger starts at 50,000 feet. While it is correct that you start to see the sky as black not blue around 60K', to "burn" on re-entry you need both potential energy (from height) and kinetic energy (rotational speed or "orbiting speed", a measure of how fast you'd be moving compared to the ground below). As the Jaeger was geosynchronous (still) above the planet, it would have a kinetic speed of 0, and would not heat on reentry, especially not from 50K'.
Throughout the movie, the control team members erroneously refer to enemy targets as "bogeys". A bogey is an unidentified target; once it has been identified as hostile, then it is referred to as a "bandit."
(at around 33 mins) When the construction of Gipsy Danger is described, it's stated that the Jaeger has solid iron armor with no alloys. This seems like an exceptionally poor choice for a vehicle expected to operate primarily in water, as unalloyed iron would corrode rapidly and require constant maintenance. In addition, alloys of iron, such as steel, were developed specifically because they provide greater durability as armor and weapons, so citing a lack of alloys as a benefit seems odd.
(at around 20 mins) When the Kaiju breaches the sea wall and is threatening Sydney, it is first shown striding under the Harbour Bridge, towards the Opera House. Travelling in this direction, the Kaiju would actually be heading towards the ocean, not coming from it.
The "evacuation pods" in the Jaeger appear to have no propulsion mechanism other than air bags that float the pod to the ocean surface. This would render the eject function useless, as pilots would only be able to escape from the Jaeger if it was completely submerged underwater.
As the final attack on the breach commences, at 1:46:00, Pentecost informs Chuck that the breach is 3000 meters down. Chuck responds "Half a mile!? I can't see a bloody inch" 3000 meters is close to two miles. It is also odd that someone with an Aussie accent would refer to imperial distances, especially in times of stress.
(at around 1h 14 mins) Gipsy Danger is said to be immune to an EMP because it is "nuclear powered" and therefore "analog". This would be nearly impossible as the two have nothing to do with one another. Most nuclear reactors have many digital parts and computers controlling them. Furthermore, the cockpit of Gipsy Danger is filled with what appear to be digital computers.
When the kaiju emits the EMP, it takes out the jagers, the command center, and presumably anything that's digital. But when the Gypsy Dancer attacks, it is carried by a fleet of helicopters, which presumably have the same digital circuits.
At various points in the movie in the control room. When members communicate to the Jaeger pilots via the microphone, they do not press the black button on the mic. The Shure gooseneck 515 series microphone requires that the black button is pressed in order to transmit. Therefore, the times characters speak into the mic without pressing the button would result in no voice transmission.
The goof item below may give away important plot points.
At the end of the movie when Mako and Raleigh's liferafts are seen from above, only Mako's liferaft is ejecting the fluorescent green marker dye when it breaks the surface of the water. Mako's raft immediately started to eject the dye as soon as it broke the waters surface while Raleigh's does not.