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 a B-29 as mentioned earlier in the film (an understandable mistake, since Jimmy Stewart commanded a B-24 Liberator during WWII, 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.
When Col.Holland is going for his first ride in the B-36, they taxi to the end of the runway. The AC calls for takeoff power and 100% power on the jets. The camera shifts to the outside of the aircraft and the pitot covers are still in place which would not happen if the aircraft is going to fly.
When Col. Holland can't move his arm when approaching Kadena, he has the co-pilot work the throttles for the landing. Earlier in the movie, he allowed Rocky to make a landing at MacDill from the back seat, so why didn't he let Leo make the landing at Kadena? Since they were on instruments anyway, it would not have been a problem for Leo to make the landing from the back seat.