The Qixi Festival that Dre attends with his mom and Mr. Han falls on the seventh day of the seventh month of the lunar calendar. Though the exact date in our modern calendar varies from year to year, it is usually sometime in August. However, a few scenes after this festival, Mr. Han says the date is June 8th. This is not possible unless 11 months had passed in those few scenes.
In various scenes Dre is said to be "speaking Chinese" (or making a mess of it). While this is technically correct, there is no universal language called "Chinese" - instead there are many dialects (none of which are mutually intelligible). The dialect native to Beijing (and spoken by the characters in the movie) is Mandarin.
The newspaper clipping on the hood of Mr Han's now-beat-up car is covered with pieces of windshield. Dre picks it up, removing the glass. When the camera shows the clipping right after, the pieces of glass are back on the clipping.
After Dre gets beaten up at the playground, you see him in the bathroom putting on his mom's foundation over his black eye. He puts the foundation on almost an inch under his eye, only covering about an inch in height on his cheek. Yet if you look closely in the next scene when he is walking to school with his mom, the foundation is now directly under his eye and covers most of his cheek.
When Dre watches the woman charming a cobra on the mountain, he first watches from behind a window with its bars all in place and parallel. Then he walks farther up the hall and peers out a second window that has one bar bent out of place, allowing him to peek his head farther out for a better view. He has his head out this second window when Han calls him, but in the next shot he's behind the first window again as he responds to the call.
On the way from Beijing Airport (North East Beijing) to the Parker's new house they drive past the CCTV building (South East Beijing) and then the Birds Nest (North Bijing). No one would drive this route.
During the celebration of winning the first rounds of the tournament, a photographer is seen taking pictures from very close range; under one meter. His camera is sporting a telephoto lens, which usually have a minimal macro distance of 1.2 to 1.4 meters. There is no way the photographer could have been taking clear pictures.
Though the plot of the movie revolves around kung fu rather than karate, the title "The Karate Kid" is a direct reference to the bullies calling him the "Karate Kid" when he takes up a stance at the beginning of the fight.
The plane shown taking off to take the Parkers to China is different from the one shown landing in China. The planes shown have different registration numbers which are seen on the fuselages (specifically B-2460 and B-2443). However there is no explicit statement that the plane we are looking at is "their" plane, just that it's an Air China plane taking off / landing.
When Mr. Han and Dre visit Master Li's Kung Fu school, they witness him punishing one of his students by slapping him in the face. The student falls backwards as if being hit almost a full second before Master Li's hand would have connected with his face.
In the alternate ending for the film there was an extended fight scene between Mr. Han and Master Li. At the end of that scene Master Li was left lying on the ground near the stage. This ending was scrapped and the existing footage re-edited into the theatrical ending. As the Fighting Dragon students bow to Mr. Han and Dre there is a cut-away to Master Li looking surprised. When it cuts back to Mr. Han and Dre, Master Li can be seen on the floor in the background between them.
In the final scene when Han and Dre bump fists after the tournament, Master Li is on the ground behind them. This is inconsistent the feature ending, where Li was standing up just moments before the fist bump. It is consistent with the alternate ending, where Han has just beaten Li in a fight and left him on the ground.