Deliberate mistake: When the investigators are in the basement wine cellar, they turn on their flashlights to illuminate the pitch black area as they walk around. However there is a light source moving with the camera shining on them as they are casting a shadow behind them.
In Which We Meet Mr. Jones - S1-E7
Deliberate mistake: Dunham comments that she works for Homeland Security. While the FBI is under the jurisdiction of the Justice Department, she works as part of a special Homeland Security task force; so saying she works for DHS is an accurate statement.
Deliberate mistake: When filming the fake movie with Sophie playing the nun, many holes are made in the door, but when she slowly kneels to the ground, there isn't a hole behind her, so it seems like she's been miraculously saved. But as the scene continues, the nun starts to bleed and dies. Technically, they couldn't safely put a squib that close to her. (00:21:15)