When Jay Baruchel and Craig Robinson are running from the dog-like demon in the neighbor's house, Craig saves Jay by pushing him out of a glass door. As Craig is jumping into Jay he is securely wearing the backpack full of food but when they crash through to the outside the very next second he is not wearing it.
When Emma Watson is trying to escape the group after overhearing the rape conversation, she hits Seth Rogen in the face with the butt of her ax, breaking his glasses in half. While he wears them with tape around the right temple and hinge for the rest of the film, the glasses clearly broke in half at the bridge, which would require tape on the bridge to hold them together.
When Jonah Hill vomits on Seth Rogen and James Franco, they are completely covered in vomit. In the next shot their hair is dry and their t-shirts are only a bit wet.
Approximately 56:23 into the movie, the camera is showing James Franco on the floor above and the backs of Danny McBride and Seth Rogen. Rogen is visibly cracking up. Yet, when the camera is showing their faces, they're straight-face.
When Seth Rogen and Jay Baruchel are in the convenience store and the rapture hits, all of the windows across the front of the store shatter in the wide shot. A second later, when the camera cuts to Jay, the windows are intact.
James Franco knows that the gun they have is loaded with blanks, but he panics with everyone else when Jonah Hill pretends he is going to kill himself with it. However, blank rounds can be deadly when fired at close enough range. In fact. In 1984, actor Jon-Erik Hexum died in a similar situation when he used a gun loaded with blanks to feign suicide as a joke.
With it being the end of the world and not having running water, they most likely wouldn't have electricity as well. But, in multiple scenes you can see lights on in the ceilings and lamps being used. And if they do have electricity, they should be saving their candles.
When taking inventory on the food and items they have after the disaster happens. Seth Rogen says "a video camera from the movie 27 Hours" and James Franco corrects him and says "127 Hours (2010)" and Seth writes down and says "127 Hours". Jay Baruchel is seen struggling to keep a straight face and covering his face.