When Holland (James Stewart) is sent to Greenland, his wife, Sally (June Allyson) asks him to bring her back a penguin. Penguins only exist in the southern hemisphere, Greenland is in the northern hemisphere.
At one point, Dutch refers to his old aircraft as a B-24, not B-29, as mentioned earlier in the film. A perfectly understandable mistake, since Jimmy Steward commanded a B-24 Liberator during WW II, flying 29 combat missions.
When the wing is being briefed about the deployment to Japan, Jimmy Stewart addresses them. Some will fly non-stop, others will refuel in Seattle. The pin on the map representing Seattle is actually placed at Portland.
While trying to resolve his i.d. problem at the air force base gate, Dutch Holland instructs the cab driver to pull over to a nearby parking lot. He then gets out and takes his bag but doesn't pay the cab driver.