Continuity mistake: After Tony gets the armor back at the Mandarin's hideout and meets up with Rhodes, they go by boat to the harbor. On the way Tony calls the Vice President - during this scene the VP's phone changes, note the camera on the back of the phone is visible, but in another shot right after, it is a completely flat back panel.
Continuity mistake: When Stark gets trapped by the water tank he is completely soaked. When he walks back to the car with the boy he is completely dry and his hair is restyled with not enough time for this to happen.
Factual error: When the Iron Patriot armor is shot at on Air Force One, the bullets ricochet and smash a window. The windows on Air Force One are bulletproof.
Other mistake: When Tony's house is destroyed by the Mandarin, watch closely when Pepper is blown through the air, just before Tony attaches his suit to her - she's very obviously wearing high-heeled shoes that would make it impossible for her feet/ankles to fit correctly into the suit without them being crunched up by the suit's boots. But later on, when she gets out, if you listen, you can hear the "clicking" of her unbroken heels as she runs around.
Character mistake: Pepper is in hospital with an IV line in her arm. A nurse comes in and injects a drug into the line. The problem is: The nurse doesn't kink the line above where she injects, which means the drug can go up the line and into the IV bag as well as going down the line. When a drug is ordered IV, the doctor who ordered the drug wants it to go in quickly, not slowly from the bag or upper tubing.
Continuity mistake: Before Tony sends a holiday greeting to the Mandarin his car is pointed one way when he gets in, then the car faces the other direction.
Continuity mistake: When Tony is giving his address to the Mandarin through the media the reporter in front of him has a cell phone which changes design between shots.
Continuity mistake: When the suit starts to arrive at the bad guy's house in bits, Tony Stark is walking down a long hallway. Scene cuts to outside and his facemask arrives, but the building he is in front of isn't big enough for the hallway we just saw.
Factual error: In the scene in Tennessee, when Tony is being chased by the glowing "homeland security" lady. He throws a jug of oil and lights it on fire with the red hot handcuffs that he takes off. Then he somehow manages to slip a set of dog tags in a microwave and turns it on. This is to create a spark to explode the gas line that he pulls from the wall, which somehow isn't ignited by the large grease fire in the same room.
Factual error: At the end Tony drives an Audi R-8 e-tron, which is an electric car, and while it has a sound system to produce an artificial engine noise, it doesn't sound like what is heard.
Plot hole: How come the Extremis serum was able to regenerate Ellen Brandt's arm but not repair the scar on her face? According to the movie, the serum is supposed rewrite the body's genetic code and constantly repair damaged cells, so her face should have been restored to its original state, just like her missing arm. Even if she had been born with the scar the serum would have healed it, just like it healed all of Aldrich Killian's physical impairments.