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: Tony is in the bedroom with Maya Hansen, Happy is seen closing the sliding door. With the door just inches from being fully closed, a quick cut to Tony and Maya kissing, then back to Happy, and the door is almost fully open, with Happy in the process of closing it again just as the plant explodes.
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.
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.
Factual error: When Maya is trying to get Killian to let Tony go, she holds a dose of Extremis to her neck and says "1200 cc's - A dose half this size, I'm dead." 1200 cc's is over 40 ounces. What she's holding is barely the size of a cell phone.
Continuity mistake: When the Mark 42 is "done" assembling for the first time, in Tony's workshop, a digital model of the suit appears in the lower right corner of the screen. In this render, the suit is complete, but a moment later there is still one piece missing, which hits Tony so hard the entire suit falls apart.
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.