While in the hospital room, Lily's shoulder strap is alternately on/off her shoulder/bandage.
When Marshall asks who would take the unused room in the apartment if he were to move out, he uses Barney as humorous possibility. He says that Barney cooks in the nude. In "World's Greatest Couple", Lilly cooks for Barney, which makes Barney mad, as this is a change from his normal lifestyle, which does not involve having food cooked at his place.
When Barney is convincing Ted that Lily moving in will mean he's being edged out, the position of the coat on the bench back behind Robin shifts between cuts, going back and forth between hanging over the edge of the back and being laid sever inches from the edge. Then when Barney leaves to sit with his date, in one cut it is completely gone, only to reappear again.
The old 'shocky' coffee maker is said to be rusty and cause electric shocks. The device shown in the episode, as all of them from its era, is made of aluminum. Aluminum does not rust. Moreover the electrical shocks story is also implausible. If, due to wear, heating coils have become exposed, pouring any liquid like coffee on them would cause a short circuit which would blow fuse or trip earth fault device. In any case, the lights would go out permanently, not just flicker.