start of the enumeration points from answer posted by papakias
Arthur being in a romantic relationship with his neighbour
His first appearance in the TV show where the presenter (R. DeNiro) hugs him
There is also another scene in the movie, which is at least hinted to be an illusion of arthur.
Arthur stoically empties his fridge at night, climbs in and shuts the door.
The scene fades out and is cut the a day-lit scene showing the fridge from a similar angle, without any of the contents on the ground, the camera then pans to show Arthur lying on his bed.
Of course it can be argued, that he climbed out of the fridge, restored its contents and then went sleeping, but it could also be a simple dream or delusion.
Psychonaut argues: The fridge he gets into is an old-style one that closes with a mechanical latch, not a magnetic seal like modern appliances. There's generally no way of opening those old fridges from the inside.
Additionally it seems that he is firing a few more bullets than he should be having in his barrel when dealing with the guys in the first train scene.
He does not seem to be in a reloading-mood in the scene. So it is either a (deliberate) mistake or something that Arthur did not seem to care about in his delusion.