In the opening scene we see David Aames awaken to the sound of his alarm clock, he quickly turns it off. But in reality that alarm clock was a JVC FS-SD1000 and the button Tom Cruise touched was the volume down button which would not have stopped the alarm nor made an indicator sound upon being touched.
Where David points to the mole on Sofia's chest, you hear her chuckle (and the closed captioning says [Chuckling]. However, her facial expression remains completely unchanged and the chuckling was clearly dubbed in.
Eye line off in final scene. When David runs to jump off the roof, he jumps up on the ledge and then stops and turns around to look at Sophia. She is looking up at him, so he should be looking down - since he's higher on the ledge - but he's also looking up.
In the closing credits, there is a credit for The Who's performance of "Baba O'Riley" from the documentary The Kids Are Alright. Actually, the short clip of The Who (seen during the montage in the final scene of the film) is footage of their performance of "Won't Get Fooled Again" from the same film.
As David returns to his car the morning after his night with Sofia, Julie approaches, having waited for him in her car down the street. How could Julie have known where to find him? Julie didn't know Sofia. Julie was still at the party when David and Sofia left. And from the setting and street sign it's clear that Sofia lives near the Manhattan Bridge (Water St.) in Brooklyn, an entirely different part of New York City than David (who lives in The Dakota across from Central Park), so Julie couldn't possibly have found him by just driving around searching. Brian could have told Julie where Sofia lives during the party, but he just met Sofia earlier that day and it would have been an odd thing to do, if he even knew Sofia's address yet himself.
When David and Brian are in the car in the beginning you can clearly see that they are about one or two feet higher compared to the other cars, even though they are in the relatively low Mustang, revealing that the car is probably on a trailer rather than on the road.
The goof items below may give away important plot points.
When David comes back to Sofia's apartment towards the end of the film. He rips off the photos from the fridge. But in the next shot when he is throwing something to the wall, the photos are still on the fridge.
When David is having his 'erased' memories explained to him in the elevator by Edmund Ventura, he his told he battled the board for control of the company. This is accompanied by a shot of the board around a table. The TV at the end of the table reflects camera crew as the shot pans.
Toward the end of the scene when Brian confronts David, saying, "Hit me, but don't ever hit a girl!" In the shots of Brian, snow is falling. When it cuts back to David, no snow is to be seen. Cameron Crowe notes in the director's commentary that this mistake was caught in post, but they excused it because this is one movie where it might actually make sense.
As David thinks he is having sex with Sofia (but it turns out to be Julie), the shots that include both Tom Cruise and Cameron Diaz show her hair to be multiple shades of blond/ash blonde with highlights. However, when the angle changes to be from behind the bed and all you can see is Tom's face and the top of Cameron's hair, it is clearly not her hair but a one color, bleach blonde wig.