(at around 1h 35 mins) When the occultist restarts the curse and wakes up Arne, the covers are on him from the waist down. Afterwards, almost every other shot, the covers are totally off him, showing his boxers and legs, and then back on him again. The scenes continue to alternate between the two scenarios for a whole minute.
(at around 11 mins) When Ed had his heart attack the doctor told then they have to put a stent into the artery. Doctors were not placing stents, much less balloon angiography in 1981. The treatment would've been major open heart by-pass surgery.
(at around 10 mins) When Ed Warren (Patrick Wilson) has a heart attack, Lorraine (Vera Farmiga) instructs someone to call 911. Considering they were in Brookville, Conn, it is unlikely 911 services were available to such a rural area at that time. Chicago suburbs finally got access to 911 in 1981 (the same year the movie took place). Moreover, at that time not even 50 percent of the U.S.A. had access to 911 services and those areas were only urban and suburban.
(at around 38 mins) Under the house a Rollei 16S Submini is used to take photos. Flash photography is shown but the Rollei 16S Submini does not have a built-in flash and no external flash was connected. (Looks like the light meter is confused for a camera flash)
(at around 28 mins) Ed says "The court accepts the existence of God every time a witness swears to tell the truth." In fact, in the United States, you are not required to swear on any religious text or to any god in order to give testimony in court. You may affirm that your testimony is truthful without taking any sort of religious oath whatsoever.
When Lorraine Warren inspects Jessica's corpse in the funeral home, the corpse's big toe slightly twitches.
(at around 18 mins) The Gene Kelly version of The Three Musketeers (1948) is shown to have been playing in the movie theater that Ed worked at on the day that he met Lorraine. This movie came out in 1948, while in real life Ed Warren and Lorraine Warren got married in 1945.
(at around 27 mins) When Ed and Lorraine are attempting to convince the lawyer to represent Arne, there are several books visible on the shelf behind them. One is "Jonathan Strange and Mr Norrell," which was published in 2004 and would be impossible to exist, more or less to even get a copy in 1981.
When Lorraine is going beneath the house she asks Warren to hold her purse, but when she finds the relic there she has her purse and takes out a camera.