During the reading of Patrick's father's will (set no later than 1929), it is stated that his residence was on Lake Shore Drive in Chicago. Lake Shore Drive was not constructed until 1937 and would not be known by that name until 1946.
When Mame, Agnes, Ito, and Patrick are preparing to dine with Beau, Mame remarks, "I never thought Santa Claus would look so much like Rhett Butler." This part of the movie is set in the early Great Depression, well before Gone with the Wind or even the book (1936) was released.
As the opening credits end, when we see the large portrait of Mame in her hallway in a wide shot, Mame's hair is flaming red's to match Lucille Ball's famous hair color. But when we see it again a minute or two later, from a closer view, the same portrait now has brown hair which matches the brunette wig Lucy is wearing in that scene.
When Mr. Babcock and Mame argue, she has her hands protectively on Patrick's shoulders. Shot cuts to Babcock saying "That's not a school, it's the Garden of Eden," and when it cuts back to the longer shot, her hands are covering Patrick's ears. This odd bit of continuity is due to cut dialogue, in which she declares "What could be more wholesome and natural?" and he responds "It is not wholesome and natural for boys and girls to run around half nude." Mame is shown covering Patrick's ears while responding "Mr. Babcock! Not in front of the B-O-Y!"
When Mame and Beauregard do their elegant soft-shoe routine in the middle of the title number, she holds her riding crop in her left hand. When the other dancers join in, the angle changes and goes to a close-up, and as it pulls back, the riding crop is now in her right hand, though her arms were linked by the dancers from one shot to the next, and she never realistically had a chance to move it.
The goof item below may give away important plot points.
Mame tells the Upsons that the pregnant Agnes Gooch is "un petite enceinte." While "enceinte" is correctly used - as a colloquial term for "pregnant" - the correct way to say it would be "un peu enceinte," i.e., a little bit pregnant. Mame, a cultured and sophisticated world traveler, would not have made such an error.