In the beginning of the story, in the kitchen, Tony is wearing oak leaves on his shoulder, indicating his grade as Major, but on his closeups, his grade changes to Captain. Cutting back to the full two-shots with Jeannie, his grade is Major. These shots were likely done later since the very next episode is when he is promoted to Major and his uniform was updated. Wardrobe should've switched the grade back to Captain's bars for these "pick up" shots.
When Nelson is first brought in to the warehouse, he is wearing his captain's bars. When the Chinese spies are injecting Nelson with truth serum, Nelson is clearly missing his captain's bars on his left shoulder. The right shoulder still has the captain's bars. The bars are still gone when he returns home. They only reappear again when Nelson enters Dr. Bellows's office at the end.
Beginning of the scene when the coffee pot is pouring the coffee into Tony's cup you can see that the prop misses the cup and spills the coffee onto the table.
The nameplate on General Peterson's desk says, "Gen. A. Peterson," but the general's first name is Martin.
If the Chinese secret agents already had truth serum and it can tell the truth why are they requesting Tony to tell them everything they can just use the serum to get all the information.
The Chinese spies appear to be speaking in Cantonese which is spoken in Hong Kong, rather than Mandarin which would be spoken in communist China.
At the end, Capt. Nelson tells Gen. Patterson and Dr. Bellows that Capt. Healey's tales about Chinese spies were from a play that the two Captains were writing. That makes no sense. The fact that there were spies willing to kidnap astronauts would have been a national security issue and it would be insane not to report them. Of course, they'd have to make sure to have Jeannie change them back to human, instead of donkey and bird, before arranging their capture.