After Ambassador Trentino has announced, he steps down the stairs followed by two gentlemen and crosses two ladies before he kisses Mrs. Teasdale's hand. Next shot the two ladies are just behind him yet and the last man after him is already on Mrs. Teasdale right hand-hand side, giving his back to the wall.
During the fight with the lemonade vendor, the vendor gets squirted with lemonade and his shirt gets soaked. Seconds after, while the vendor is tussling with Chicolini again, we see that the vendor's shirt is completely dry.
When Firefly arrives at his reception, he is wearing a dark swallowtail coat. Then after Mrs. Teasdale greets him, we go to a two shot, and Firefly is now wearing a lighter shade jacket with dark piping and a glove in the breast pocket. When Ambassador Trentino enters, Firefly is again wearing the swallowtail coat, which he retains until the very last shot of that scene, when he is again wearing the lighter jacket, without the glove in the pocket.
When they meet, Mrs. Teasdale holds Firefly's right hand while he holds the cigar with his left hand. The subsequent shot shows him holding the cigar with the right hand and the deck with the left one.
During the "Going to War" sequence, the four leads are shown kneeling down. In the wider angle shot showing the entire chamber, Bob is shown standing up while the other three are still kneeling down. In the next shot facing the four from a lower angle, Bob is kneeling down again.
While this may not be a "goof" but only an example of the general zaniness of the film, the scene in which Firefly is first presented as the new head of state is supposed to take place in the morning (the gag of Groucho being woken at just the right moment by his alarm clock and having to slide down a fire pole to arrive -- late -- at the reception, the call for "His Excellency's car" with Harpo driving off and leaving Groucho stranded in the sidecar, taking place in broad daylight, etc.). Yet all of the attendees at the presentation reception are clearly in formal evening dress (the women are wearing floor-length evening gowns, the non-military men, including Ambassador Trentino, are in black tie and "tails", etc.). Perhaps ironically, only Groucho himself is -- appropriately -- wearing a swallow-tailed morning coat (albeit one that looks like it hasn't seen the dry cleaner's for some time).
The removed horseshoes that we see lying on the floor (from the horse that Pinky was riding) are not actual horseshoes but the type that are used in the game of horseshoes. They are much larger than regular horseshoes.