There are numerous changes in Delacourt's dialog that are clearly visible after Max and Frey are on Elysium. She says "Get these two out of here." while looking at Frey and her daughter but she clearly says something else. You can hear the difference between the lines that are changed and those that aren't.
SECDEF Delacourt is wearing a white-on-white suit at the pool party when she receives the communication about an illegal ship approaching. In the next scene, she is walking to a control room wearing a metallic gray suit. The transit time from Earth is cited as 19 minutes, so it is unlikely she would have changed so fast. From the control room, she is urgently summoned by the President and appears in front of him wearing the white suit again.
While Krueger receives the message about being discharged, he cooks some meat using his sword. While doing so he has his special armor glove on and the glove is connected to the argumentation on his chest. During the next scenes the glove is sometimes connected and sometimes not.
While the CEO was programming the new boot sequence we see him changing assembler commands. In the next scene we see the translucent screen from behind and the content of the screen has apparently changed.
The original "Stanford Torus" design of a wheel-shaped space colony had an offset mirror angled to reflect sunlight onto mirrors set around the central hub and then outward to the ring. Elysium has no such arrangement and sunlight simply shines directly to the interior of the ring. This is completely impractical, as Elysium rotates (about every 6 minutes). This means that location around the habitat would have an ongoing series of periods of "day" and "night", each lasting only 3 minutes, which would be extremely disconcerting for the inhabitants. In addition, Elysium appears to have its rotational axis pointing towards the Earth, and as it orbits the Earth that would mean that (a) it rotates with respect to the Sun, and so the amount of sunlight and its angle will be continuously varying, and (b) during each orbit it will spend time in the Earth's shadow and not receive any sunlight at all.
When the little girl with crutches is smuggled onto Elysium, the medical scanner diagnoses "multiple compound fractures". However, a compound fracture (modernly called an open fracture) is where broken bones cut through the skin. There were no such injuries on the girl as she walked with her crutches.
A Stanford torus, such as Elysium, would be too small to be visible in the sky of Los Angeles to the naked eye in the daylight. Comets and asteroids which are orders of magnitude brighter than Elysium could ever be, are not visible in space to unaided human sight.
When Max and Julio's car is attacked by the delayed-detonation RPG's launched from Kruger's ship, the attack angle and detonation are contradictory to the laws of physics. Since Max and Julio were traveling backwards, the explosive force of the two devices would have lifted the car from the front end backwards, not from the rear end forwards as shown.
After being struck by Kruger's missile, Max escapes his overturned car by gripping the door with his fingers and bending it open. Max's fingers are not augmented like the rest of his body. He would not have been able to grip the door as shown and bend it open without severe injury to his hand.
When the illegal shuttles are heading for Elysium, shots inside the ships show beer bottles floating. This would only happen in free fall, but all shots of the exterior show the engines running. In that case, acceleration forces would cause loose objects to "fall" in a direction opposite of the acceleration.
When the CEO is changing the boot sequence he first of all modifies a 64bit ASM program, which might is outdated in 2154. Second the sequence he is adding is reading the control register into EBX (MOV EBX, CR0) followed by using an OR to modify the EBX register. There is no writing the result back into the control register. After that the copy routine of the interrupt table overwrites the EBX register so the changes to the CR0 is clearly lost at that point.
Therefore the displayed changes have no effect.
When the CEO compiles the new boot sequence from his changes it took about 2 seconds and the final line does say: 6028 modules processed. 9357 lps. LPS means lines per second and since he was using an assembler it would say that at max he was dealing with a program about 18000 commands which is quite small for a boot sequence.
While the army vehicle hovers over the dirt after the CEO was shot down, the dust is moved by whirling air. The movement matches those of a helicopter/rotor but is not matching a jet/propulsion engine that is pointed towards the ground.
When Max is transported to Elysium there is a scene where the shuttle is turned upward, goes full throttle and leaves earth orbit. In the next scene we see inside the shuttle and Max holding his granate and talks to the woman without being fixated in his chair. In the whole movie there is no technology used to compensate for this kind of force. It is even so that in later scenes the vehicle inmates experience the force of acceleration.
The craft shown flying from Earth to Elysium appear to be solely air-breathing vehicles and could not possibly reach orbit. Even if they had some rocket propulsion that was not obviously visible, they are not large enough to provide sufficient power for a journey to Elysium.
While no current shoulder-fired weapon is able to achieve escape velocity and overtake a traveling spacecraft, canonical sources list the primary purpose of the "4Sure" ballistic missile launcher used by Kruger as engagement of targets in earth orbit.
The missile is stated to travel 518 km/hr, or 45 times earth escape velocity. The effective range is described as in excess of 15,000 miles, therefore the weapon is more than capable of intercepting the illegal spacecraft as depicted.
When Max gets trapped in the droid chamber and suffers a lethal dose of radiation, the procedure stops immediately when the system detects organic tissue in the chamber. Even by today's strict industrial security standards, the human worker should have been recognized BEFORE the chamber was fired up and the sequence would not have initiated at all.
Elysium was shown several times not to have a solid roof. However, this would mean that, despite the rather slow rotation imparted to create the effects of gravity, the atmosphere would have very quickly escaped to space. Some physical barrier would have been needed to make the station viable.
Max wearing his exosuit performs heavy muscle activities clearly indicate that he is not supported by his exosuit at all. The scene where he pulls open the lock of the chair in the crashed vehicle he hangs with a single hand in a bowed position clearly using only his muscle strength to hold him in position.
The goof item below may give away important plot points.
During the Elysium computer reboot, a present era BIOS POST is shown amusingly featuring IDE with master and slave channels. This would already be considered obsolete and rare in the movie's release year of 2013, due to its widespread replacement with the SATA bus.