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.
Tony and Killian meet in 1998, along with Pepper and 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.
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.
When the DHS agent working for Killian who came to get the file 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.
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.)
When Tony is zip-tied to the bed, he asks one of the guards the distance between Miami and Tennessee. The guard says 832 miles, which is correct for some parts of Tennessee. This means Tony's armor is ~832 miles away. Even assuming his armor travels at 1000 miles/hour, it would take his armor almost an hour to get to him, rather than the few minutes shown in the movie. Tony's armor would have to travel at almost 10,000 miles/hour to get to Tony in 5 minutes. The current airspeed record for a plane is only 2,193 miles/hour. Tony's armor, at the current airspeed record, would still take 23 minutes to get to him.
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.
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, 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?