Visible crew/equipment: When Peter stands up after being bitten by the spider, there's the reflection of the cameraman with headphones on the television set behind him. (00:10:00)
Visible crew/equipment: When Peter wakes up and discovers he can't see through his glasses, you can see a phantom hand on the left sliding out of view in the mirror. This happens the last time Peter tries to see through his glasses. It's not a reflection of his hand due to the angled edge - it's on the main part of the mirror. (00:18:05)
Visible crew/equipment: When Mary Jane is on the balcony at the Oscorp Festival and the Green Goblin causes part of it to start falling apart, the wires and chords causing the balcony to fall down and pull back up can be seen. (01:07:20)
Visible crew/equipment: When the fire breaks out in the building, as the firetruck goes by you can see the reflection of the camera crew on the truck, including a man wearing a white shirt. (01:22:00)
Visible crew/equipment: At Thanksgiving dinner, when Aunt May looks outside when she sees something, the reflection on her glasses shows what appears to be a green screen. (01:26:00)
Visible crew/equipment: In the school fight scene where Peter beats Flash by punching him and making him collide against a teacher, his tray falls on his face. In that precise moment a camera with a hand on it can be briefly seen on the left side of the screen just a few moments before the shot points downwards.
Answer: Remember that Green Goblin (Norman Osborne) just found out that Peter Parker is Spider-Man. Aunt May's comment about everyone knows he loves Mary-Jane concerns him because he now realizes that Green Goblin might find/kidnap Mary-Jane just to lure Spider-Man into a fight. This is the basis for the whole reason he can't be with her, as he would be potentially putting her in harm's way all the time if they got together.
Jazetopher