Corrected entry: After Serenity crash lands on Mr. Universe's planet, Kaylee says she can rig the door so it won't open once it's shut. All points say she has done this. However, once Mal comes back, he asks where River is. The door which is supposed to be shut for good opens dramatically to show River standing in the middle of dead Reavers just in time for the wall to fall down. River is nowhere near the door, and no one inside the room has gone anywhere near it either, so regardless of just how permanently that door was sealed, nothing's caused it to open.
Corrected entry: In the pilot of Firefly, River seems shocked to see Simon when she wakes up from cryosleep. In Serenity, she sees Simon before going into cryosleep, so she should remember. One could argue that due to her mental problems she simply forgot, but River's memory has always been extraordinary, even in terrible circumstances.
Corrected entry: According to Joss Whedon, people in the Firefly universe speak only English and/or Chinese (see Special Features and the movie's subtitles), and yet, the writing we see everywhere is Japanese (there's a difference between Japanese and Chinese script; it's obvious for those who know).
Corrected entry: During the bank robbery scene, down in the vault. Just after Jayne shows up to warn Mal of the reavers, count the people moving behind Mal. Zoe was already in the vault and Jayne went to join her. Then Jayne runs behind Mal out of the vault, followed by someone else (I think it was Zoe) and then Zoe again. Though it's unclear, three people exit behind Mal after only two went in.
Corrected entry: In the scene near the beginning when the Operative is investigating the escape of Simon and River, he is reviewing the recording of them escaping from the facility. During his conversation with the doctor, he changes the recording to show the doctor's and Simon's conversation, pointing out the phrase "Key members of Parliament." Later, after he has killed the researcher, the Operative goes back to the recording and it's back to Simon and River escaping, seemingly without him changing it back.
Corrected entry: After the reavers first show up in town, there is a harrowing chase back to the ship. At the finale, it's Mal and friends heading straight for the reavers. Serenity shows up in the middle of them with the cargo doors open. The crew slides neatly into the ship and to safety. A few moments later, a reaver, along with the top of his ship enters through the same door. The mistake here is that the reavers were going the opposite direction. Not only would the reaver ship just scrape the bottom, it would have had to instantaneously change directions in order for the top to be shorn off like that. Basically, for what happened to work, the reavers would have had to have been directly behind them instead of head on, as they were.
Corrected entry: When the Operative is reviewing the computer file on Mal, one of the notes reads "Awarded Valor Commendation: Battle of Serenity Valley." Since the Battle of Serenity Valley was the decisive battle of the war, which the Browncoats lost, who awarded him the commendation? The Browncoats had surrendered, so they were certainly in no position to hand out medals, and the Alliance would surely not have given Mal an award for fighting them so tenaciously.