Factual error: When August sits down at the piano in the church and begins to play, he plays a passage of ascending notes (his hand is actually on camera so you see it move to the right, up the piano). However, the audio that you actually hear is a passage of descending notes.
Deliberate mistake: When Evan/August first meets Arthur at the park, after playing, you see Arthur leaving his spot without taking the cable and (presumably) the amp as well.
Audio problem: When Evan plays the guitar the first time, you can hear some chords and notes that aren't actually being played at all.
Plot hole: When Mr. Jeffries finds out that Evan is named August Rush too, he goes to the Central Park concert and sees Evan onstage, however he couldn't know that Evan was having a concert, or even that Evan was at that place at that time.
Other mistake: On Evan's missing poster that Wizard finds in the park, it states his age as 12, but that he's been missing since Feb 2006. Given that his date of birth is December 1995, this would make him only just 10 years old. (00:53:28)
Other mistake: When Evan is conducting the orchestra, his conducting pattern does not line up with the downbeat of one as the camera angle changes.
Plot hole: 12 years have passed through the entire movie since the beginning where Lewis and Lyla first meet to the end when they together discover their now 11-year old son. However, neither Lewis nor Lyla look like they aged whatsoever in 12 years.