When Sarah is giving Joseph a shot, she gives it in the wrong spot. The needle is the size for an adult intramuscular injection which would be given in the muscular part of the arm. She gave it just a little above the elbow, which is not a very muscular region. It is possible that she would have to use the over-sized needle because of a lack of supplies in the area she was in, but she would never have given the injection in that location if she were medically trained.
Father Gionetti gives Father Merrin a copy of the Rituale Romanum (Roman Ritual). (The administration of exorcism is in Title X of the Rituale Romanum.) The title of the book embossed on its leather cover is "Roman Ritual", in English. Later in the film, Father Merrin asks Joseph to help him by reading passages of the book, and we are shown a couple of printed pages, the text of which is also in English. The Rituale Romanum would have been printed in Latin in 1949, when the film is set, because vulgar translations of liturgical texts were not authorized until the promulgation of the Sacrosanctum Concilium (Constitution on the Sacred Liturgy) by Pope Paul VI on 4 December 1963, as part of the Second Vatican Council.
In 1949, Kenya would have been pronounced "Keen-ya" instead of "Kehn-ya," as it is today. The pronunciation change came about in 1963 when Kenya became independent from British rule. Characters in the movie use the modern pronunciation instead of the old.
When Merrin remembers WWII German soldiers marching with their typical "Knobelbecher" boots, they can be seen having modern soles. This is probably due to the fact that many film productions use the almost identical East German "Knobelbecher" boots, which became available cheaply and in abundance after the fall of the German Democratic Republic in 1990.
When Merrin and Joseph perform the exorcism, Joseph responds "And also with you." This loose translation of the Latin phrase "Et cum spiritu tuo" ("And with your spirit," or traditionally, "And with thy spirit") was only coined in 1973, when the International Commission on English in the Liturgy (ICEL) released an English translation of Latin ritual texts used in the Catholic Church.
The crucifix hanging upside-down underneath the church resembles Western, Renaissance portrayals more than actual Byzantine depictions of the crucified Christ. In fact, it is even unlikely that a 5th-century church would have had such a crucifix (particularly a life-sized one like in the movie). Firstly, Christian artists were only beginning to portray Christ regularly in such a manner at that time, and when they did, he is not shown as suffering or dead as in later depictions (and the prop in the movie), but triumphantly alive and erect. (In fact, in some contemporary artworks - particularly Byzantine ones - he is even shown clothed with a sleeveless long robe known as a colobium.) Secondly, the Byzantines inherited early Christian distrust of monumental sculpture in religious art and so usually produced only reliefs: a crucifix featuring a fully-sculpted depiction of Christ crucified would have been quite odd.
The 5th century Byzantine priest at the beginning of the films has a Western-style rosary clasped in his hand. While the use of prayer ropes (alongside other methods of counting prayers) was already known around that time, rosaries only reached their present form during the 15th-16th centuries. Note also that actual Byzantine prayer ropes - known as a komboskini or a chotki - are quite different in form and function from Western rosaries.
When Father Francis takes Joseph into the church, he says "In nomine patris, et filii, et spiritus sanctu". The actual Latin is "In nomine patris, et filii, et spiritus sancti" - "In the name of the father, the son, and the holy ghost".
When the Byzantine Knight/Priest crosses himself while standing among all of the bodies (in one of the early scenes) he does so in the Roman Catholic fashion (left to right). Any Byzantine of that era would adhere to the Eastern Orthodox variant (right to left).
The machine gun that the British Soldiers set up and use against the African tribe is a German MG42. Although the movie is set post WW2, it is very unlikely that the British Army would have recycled this weapon into their artillery.
When Merrin enters the church for the first time, his guide hands him a lantern when he steps off the ladder, but after Merrin walks forward, the guide is still holding a lantern. He may have had two, but later, when the second priest comes down, the guide no longer has a lantern.
(at around 1h 12 mins) Immediately after major Granville shoots someone, the camera cuts back to show Father Merrin shouting "No!". In couple of these shots, this character appears as if the weather is raining on him. Not everyone in the shot, just him (left hand side of frame). This occurs at least twice, and for everyone else it's a sunny day. (Freeze-framing confirms the streaks of water.)
Right in the beginning theres a establishing shot subtitled Cairo. In the background you can see the pyramids of Gizeh looming over the city. These pyramids are only about 80 meters high and more than 15 kilometers from Cairo. There's no way to see them anywhere from Cairo at all.
When Merrin looks at Bession's drawing as he is walking into the church, he has it the right way up (with the cross inverted). A few moments later he looks at the hanging cross and realizes he's made a mistake in interpreting the drawing. He looks at the drawing again (now upside down, with the cross the right way up) and turns it around, giving him his epiphany about the hidden passage.
When Sarah walks through the hospital with the candle, the light emitted by the candle varies extremely. Then when the light and radio go on, one can clearly see that the candle is just a replica made of plastic.
Sarah's hypodermic needle is a lot longer before she injects the boy than it is after she pulls it out. (A trick needle slides back into the chamber of the syringe, instead of into the patient's arm. If it doesn't work properly, then the needle will be shorter after the injection than before.)
When Sarah walks trough the hospital with the candle the light emitted by the candle varies extremely. Then when the light and radio goes on one can clearly see that the candle is just a replica made of plastic.
In the final shot of the movie, as the priest is walking toward the Vatican, his shadow on the cobblestone falls at a different angle from those on the buildings. According to the background, the sunlight is obviously coming from behind us, but the priest's shadow goes off to the right.