When Chico (the dog) is lowered in the basket, the stamp in the frame is facing up. At one point in the air duct, it shows it pointing down (evident by the backing frame stand). When he returns to the room, the stamp in the frame is clearly facing up. However, after he jumps out of the air duct and starts across the room, the back of the frame is now facing up.
After the mission briefing tape has played, there are holes under the take-up reel that were not there while the tape was playing. Presumably these are for the tape destruct smoke effect.
When the phone in the Prado residence rings and later when it is used to dial out, none of the line buttons light up. These phones have five buttons that light up to indicate if a line is in use, ringing or on hold.
A bunker 25 feet underground would likely not need cooled air, but likely would need heat.
During the Dossier Scene, Jim Phelps selects Chico - an intelligent and well-trained dog - as one of the IMF agents for the mission. Chico's picture is shown on the front cover of the February 1970 issue of "The World of DOGS" magazine; however, the month is misspelled as "FEBUARY" on the front cover of the magazine.
In choosing Chico, Phelps looks at a cover from the February issue of "World of Dogs". One of the other stories on the cover concerns the "Buckminster" dog show, an obvious reference to the real world Westminster Kennel Club show. But the Westminster show is held in February, so a "February" dated magazine would have been printed a month or more before the event. The only results it could carry would be a year old.
When Chico is taking the chess piece, Barney tells him to stay. The trainers hand is briefly and barely visible moving (right of screen) to direct the dog.