(at around 1h 35 mins) Radical differences between altitudes when Iron Man is saving the airline passengers. At one point Jarvis calls out "18,000" feet and then only eight seconds later he calls out "10,000" feet. Nothing falls at 1,000 miles per hour from natural gravity. Then Jarvis calls out the last 9,400 feet in twenty seconds, nearly three times longer than it took the first distance.
(at around 30 mins) When Jarvis and Tony reconstruct the bombing of the Chinese Theatre, Jarvis mentions that, due to the extreme heat of the explosion, everything in a wide radius instantly vaporized. Yet the dog tags of the bomber, while being bent and blued, still have their rubber wrapper.
(at around 29 mins) When Stark is talking to the press and gives out his address to the Mandarin, the image on the reporter's phone is the same as from our point of view, and not what the phone's camera would have seen.
Tony and Killian meet in 1998, along with Happy in Switzerland. Fifteen years passes by and there are no significant age-related physical changes in their appearance, a nicer suit of clothes and a different hairstyle by Killian notwithstanding.
Hair lost or burnt off by those injected with Extremis grows back along with missing body parts. While hair is living within the follicle, once it's outside the skin, it's dead and cannot be regenerated in its current state. (As long as the follicle is not damaged, it can be regrown, but styling would not regenerate.)
(at around 1h 40 mins) When Tony and Rhodey are at the docks and taking cover, Tony asks for more ammo from his pistol, even though Rhodey tells him their bullets are not compatible, which Tony insists that they are. While Tony may not have any military training like Rhodey does, he is a notorious former weapons designer, and should know that certain pistols can only use certain caliber bullets.
(at around 1h 5 mins) When Tony is in the news van he is using a laptop. On the screen the current and new IP addresses are listed as 93.455.43.232 and 78.397.17.610. The IPv4 addressing system uses 32 bit addresses so the maximum value is 255.255.255.255. Also, the van has a satellite dish. Tony asks Gary to boost the ISDNs. ISDN uses the copper telephone network not satellite dishes. Also, ISDN PRI has a max downstream bandwidth of 1.544Mbps. The speed test shows a speed of over 9Mbps.
(at around 8 mins) The films opens with a terrorist attack on Ali Al Salem Air Base, Kuwait, killing the spouses and children of American soldiers at the base. Like almost all U.S. bases in the Middle East, assignments to Ali Al Salem AB are temporary rotations without family members, specifically due to the high risk of terrorism.
(at around 53 mins) When Brandt arrests Tony and the Sheriff intervenes and asks her what was happening, she identifies herself as a Homeland Security Special Agent and shows him her badge. However the badge she showed the Sheriff wasn't a DHS badge, it was a DEA, Drug Enforcement Administration, badge.
(at around 55 mins) In Tennessee, when Tony dives into the kitchen he sets a trap by putting dog tags in the microwave to spark and ignite the gas line he severed. However, within a few feet of the microwave are the open flames from Tony kicking the super-heated hand cuffs into the cooking oil.
(at around 31 mins) When Tony is observing the map of the U.S. in the wake of the Mandarin's attack on the Chinese Theater, Jarvis brings up the map of all heat signatures in the country. Tony then asks Jarvis to take away everywhere there's been a Mandarin attack. 20-30 heat signatures disappear from the map. Earlier in the movie (at around 10 mins), Rhodey tells Tony that there have been 9 total attacks, and 3 that the public know about. Where did all the heat signatures come from?