When the team is in the submarine towards the end of the movie Steve's hands are on the steering wheel. As the neon fish swim towards the sub (from the crew perspective) you can see the steering wheel without any hands on it, but when the jump to Steve, he hands remain on the wheel.
As Steve is leaving Hennessey's ship, you see the shot from inside, the door is open and though you can't see out, light coming in, indicating that it is nice and sunny out. As Steve leaves, you see outside, it is raining and there is no light to shine inside the vessel.
When Steve pulls his gun on Jane during the interview, there is an audible thud as if he'd pulled the gun and then lowered it to touch the table, however when the shot changes to face him, he is holding the gun well above the table's surface.
When Steve and Ned are in the helicopter, they're supposedly far out in remote "unprotected waters". There's a two-masted sailboat visible over Steve's right shoulder in several shots. In the same sequence, the water beneath the helicopter is shallow and churning with brown sand, indicating this was shot very near a beach.
Steve Zissou demonstrates diving helmets with integrated radio receivers and headphones in order to listen to music under water. As a marine scientist, he would know that radio waves do not propagate in sea water (except very low and extremely low frequencies, both of which need huge antennas to send messages). Given the nature of this film, these diving helmets are due to artistic license.
During the film festival Q&A at the beginning of the movie, Steve Zissou needs a translator to understand questions asked of him in Italian. Shortly afterward, while in the adventure club with Ned, Steve can perfectly understand men gossiping about him in Italian from the next room.
The "Belafonte" model is shown with an observation dome built onto the bow of the ship. It is also mentioned by Zissou. When the ships bow is exposed cresting a wave, there is no evidence of any structure below the waterline at the bow.
The goof item below may give away important plot points.
The team's Glock automatics frequently turn into Berettas when being fired. This is very noticeable during the rescue of Hennessy during the card game, when Zissou draws a Glock, but in the next shot he is clearly firing a Beretta Model 92.
After Hennessey and the Stooge are rescued from the pirates, and Team Zissou are in the midst of a shoot-out behind an old jeep, Hennessey is clearly wearing flip-flops. However, when the team reach the boat, he has gained a pair of brown boots.
Steve peels the tape off Bill Ubell's mouth when he rescues him from the pirates, and several days' growth of beard is plainly visible. Moments later, when they are running toward the beach and they reach the boy cooking lobster over a small fire, Bill's beard is noticeably thinner.