David Marcus says that he encoded the Genesis torpedo with a four minute countdown. But when Khan activates the Genesis device it begins counting down from 999 at a rate of about 2 each second, which makes it approximately eight minutes until detonation.
During the docking with the Enterprise, shuttle control gives them permission to dock at the torpedo launching bay. Instead, the shuttle is clearly shown to dock near main engineering. In addition, Kirk and Co. clearly board the ship through the torpedo bay. The shuttlepod docking sequence, as well as additional visuals of Enterprise in and departing from drydock, are reused footage from Star Trek: The Motion Picture (1979).
Several monitors on the bridge of the USS Reliant show ship diagrams that don't match the actual shape of the ship as we see it elsewhere in the movie - a result of the bridge set being used for both the Reliant and the Enterprise. Indeed, at least one of the diagrams is for a third design.
Khan has a warning torpedo fired at the Enterprise. After the torpedo explodes we see David and Carol Marcus, and McCoy, in the sickbay reacting to the explosion. Only six seconds pass (of movie time as well as "real" time) before the turbolift doors open on the bridge with David as a passenger. In the Director's Edition DVD, restored footage allowed a longer period of time between Khan's "warning shot" and David's appearance on the Bridge a moment later.
When Chekov and Terrell explore Ceti Alpha V, earphones can clearly be seen through the visors of their spacesuit helmets. When they enter the shelter they discovered and take off their helmets, the earphones are gone.
During the first battle, the blasts from the Enterprise's phasers hit the blue dome at the rear of Reliant and it explodes. In an aft shot of Reliant moments later there is a slight shot of the dome still intact, though not brightly lit.
The Enterprise is en route to to Regula I while trying to raise them on communications channels. The shot of the operations center at Regula I shows no one there, yet when the Enterprise arrives and the boarding party searches, there are bodies strewn about and one hanging by the legs.
During many scenes of the Reliant in space, stars are visible through the Reliant's dark warp engines. Most noticeable when the ship flies past the camera after Kirk says "We don't have a few minutes!" This is seen as a goof, but is actually a limitation of the blue screen shots movie makers faced in the seventies and eighties.
In an overhead shot of Space Station Regula 1 over a moon (directly after the Enterprise departs from drydock), Regula 1 is noticeably transparent; craters can be easily seen through the station. This is seen as a goof, but is actually a limitation of the blue screen shots movie makers faced in the seventies and eighties.
In Star Trek: Space Seed (1967), Khan was marooned on Ceti Alpha V with a few dozen of his followers, all of whom were about his same age (roughly their mid-thirties). Now in the movie, set fifteen years later, Khan has aged, but he's surrounded by followers who are all appear to be in their twenties; they're too young to have been the disciples with him in the original episode, but too old to have been born on the planet. However, this is possibly a sign of their augmented, genetic aging process.
When Khan leans forward, the wound on his chest "crinkles" visibly, as only a glued-on rubber prosthetic would do. However, his wound seems to be fresh and/or infected - causing the skin to swell. If this wound is now compressed by the surrounding skin/muscle tissue, it will wrinkle as seen on screen.
Khan says to Chekov, "And you - I never forget a face. Chekov, isn't it?". Although Chekov was not a bridge officer in the TV show that first featured Khan, it should be remembered that when Khan first took over Enterprise, he started with the engineering deck. Chekov was engineering ensign at the time, according to the movie's novelization.
Kirk seems to order "Phasers starboard," and the Enterprise turns to starboard, then opens fire with the port phasers. In fact, Kirk says "evasive starboard," which is confirmed by the novel and screenplay (the DVD subtitles are in error). When Kirk says, "Fire!" THEN the port phasers are shown firing.
The date at the beginning of this movie is given as Stardate 8130.4. When Kirk, Spock, and McCoy are viewing Dr. Marcus' proposal for the Genesis Project, the stardate given for the proposal is Stardate 7130.4, to which Kirk mentions the proposal was made a year ago. The first Stardate given at the beginning of Star Trek: The Motion Picture (1979) is Stardate 7411.4 (by the computer at the Epsilon IX Space Station), and that movie was to have taken place approximately 14 years before this movie (2285). The erroneous Stardates would imply that the first movie took place sometime after Dr. Marcus' proposal to Starfleet and would've been less than a year ago, which is not the case.
Shortly after the Enterprise and Reliant enter the Nebula, there is an overhead shot of the Enterprise with Reliant down below in the distance. During this shot, the rear tip of the Enterprise's starboard nacelle disappears for a brief moment.
In the computer image of the Genesis probe approaching the planet, the camera is about to collide with a mountain when a narrow canyon blinks into existence to let it through. During a talk at SIGGRAPH shortly after the movie's release, one of the animators of the clip showed the clip and explained that (at one hour/frame being rendered on a VAX 11/780), they'd rendered every 64th frame of their fractal landscape, checked for obvious issues, then rendered every 32nd frame that was not already rendered. They continued down to every 16th, 8th, 4th, and then every remaining frame. It was only at that point that they discovered that four frames were inside a mountain, and they didn't have the time to rerender with a different trajectory or a different initial landscape, so they papered over the mistake with the magically-appearing valley.
In the Kobayashi Maru scene, the images the Klingon ships on the viewer are from Star Trek: The Motion Picture (1979). The image of the Klingon Ships firing are from when the Klingon Ships were firing on V'Ger.
When Bones is on Regula space station, he gets startled by a rat, as he is paying attention to the rat scurrying away, he walks backwards through a door. Once the door closes, he turns around and is startled again because he bumps into a dangling hand - the hand moves in response.
The goof items below may give away important plot points.
When Kirk is lowering the dead scientists in the Regula Lab, he has a rope going around his waist but yet you can hear a squeaky pulley. The rope is only making contact with Kirk and the platform edge. The rope rubbing on the platform would not make the sound of a squeaky pulley.
As McCoy and Scotty are preventing Kirk from entering the irradiated chamber with Spock ("You'll flood the whole compartment!"), when the camera is on Kirk from the front, they're facing in the same direction as Kirk, but when the shot switches to a POV behind Kirk, they're both facing the other way.
When Scotty's nephew, Peter Preston, dies, he leaves a blood mark on the flap of Kirk's jacket. When you see Kirk on the bridge of the Enterprise, the blood mark is lower on the flap and considerably smaller.
If one looks carefully at the torpedo bay during the funeral scene, one can tell that the funeral scene actually takes place in the port side torpedo bay, which was destroyed in the battle in the Mutara Nebula.
Saavik, a Vulcan, is clearly crying at Spock's funeral. However, in the accepted canon, it is established that Saavik is of half Vulcan/Romulan ancestry, and Vulcans mixed with other races often produce occasional emotional responses, like Spock shows on occasion. Therefore, her tears are not a goof.