Deliberate mistake: When the Americans are in the U-571 battling the resupply sub, Tank gets the engines running and starts the sub moving forward. There is an exterior shot of the sub's propellers, with the port prop turning counterclockwise and the starboard prop turning clockwise. Then Tyler wants to turn right to heading 010, and he orders "Starboard back full", which should reverse the direction of the starboard propeller to help turn the sub. Two exterior shots later, however, the props are shown still turning exactly the same way they were before Tank reversed the starboard engine. (00:52:00 - 00:53:35)
Deliberate mistake: In the scene near the end of the movie when Trigger is in the bilge trying to stop the air leak, the concussion of an artillery shell fired from the destroyer shakes some pipes loose. The mistake is that one of the flanges visible on a section of pipe has no bolt holes, making it impossible to join that pipe to anything.
Answer: Technically he's not getting fired, but Dahlgren did not recommend Tyler for his own command, and he won't get a promotion without Dahlgren's recommendation. Therefore, Tyler's only choices are to keep being an executive officer (which is possible but not what he wants, not to mention Tyler feels betrayed by Dahlgren and isn't sure he wants to continue working with him), or leave the navy entirely.
Aerinah