When Capt. Harlock reports to Adm. Spewter after Henry volunteers his services to the Navy, Spewter says "I'm going to have a hard enough time explaining this to the Pentagon." Henry volunteered shortly after hearing the news of the attack on Pearl Harbor in December, 1941. The Pentagon was still under construction at that time.
When Henry's wife and friend are trying to save him, several people have gathered to watch the action. A young woman in this small crowd has a 1960s "flip" hairdo, yet it had been established earlier that the story takes place during World War II.
When Henry first encounters the destroyer, the number is 752. DD-752 is the U.S.S. Alfred A. Cunningham. Construction was not started on the Cunningham until February 23, 1944, and wasn't launched until August 3, 1944, two and a half years after the time frame of the movie (just after Dec. 7, 1941).
The Nazi Admiral and naval staff are all wearing US Navy uniforms (complete with rank strips and stars signifying their status as US officers of the line) but with swastikas on their caps and Maltese crosses in place of neckties.
(at around 10 mins) The shot of the people on the yacht is first played in reverse, then after the guy taps on the paper, it plays forward. You'll notice the people in the background on the right (behind the blonde in the red top) first walking backwards, then after the paper tapping, they're walking forward.
Right at the start, the admiral and lieutenant are striding purposefully through the corridors of the Pentagon. The second set of doors they pass through is marked "Restricted Area" & "Secret". The doors are opened by two marines to reveal a long lighted corridor; this is shown to be a painting as the officers walk through the doorway but then go left instead of straight on; as the doors close behind them, the whole corridor dims briefly as they walk in front of a light.