(at around 9 mins) When Sister Irene is teaching children at the beginning of the story, she affectionately teases a child by rubbing a spot of red paint on the girl's nose. She is interrupted by someone entering the room, and when the child with the red-painted nose turns to see who's come in, the red paint is no longer there.
(at around 17 mins) When Frenchie, Father Burke, and Sister Irene first arrive at the cloister, they go to the ice house. Frenchie opens the door and invites Father Burke to enter, but the priest, perhaps a bit leery of the building, makes a grand gesture of "After you" to Frenchie, who ends up entering first. The next shot shows the trio immediately after entering, walking one behind the other, down the staircase, but the priest, not Frenchie, is in the lead in this shot.
Romania was defeated during WWII and, following the overthrow of its monarchy, experienced a decade of instability while Communist forces consolidated their hold on power and fought petty squabbles among themselves. This was common news at the time and certainly would have been known by the Vatican. Even if a priest and a novitiate had been allowed into the country, they would have received an escort by a government functionary and would not have been allowed to travel alone in the country.
In 1952, Catholicism would have been under persecution from the Communists in Romania.
(at around 29 mins) The residential building has a working radio, yet there seems to be no electricity anywhere in the convent castle since only candles and oil lamps are used for lighting. Battery radios, sometimes called "farm radios", were available from the first introduction of the radio since many early households were not wired for electricity.
(at around 41 mins) When Father Burke is trapped in the coffin, he first has a flashlight and later a lighter. The scene shows his flashlight dying and him replacing it with a lighter from his pocket.
(at around 11 mins) When Father Burke and Sister Irene arrive at the house of Maurice, you can see a white light switch on the outer wall of the house. This is highly unlikely because houses in most villages in Romania in 1952 did not have electricity and relied on petrol lamps. National electrification of villages began in the '60s.
Sister Irene has clear fingernail polish on throughout the film. A novice nun would not wear this as it would be viewed as being a "vanity."
Around 1:17, the Father gives the holy relic to Sister Irene saying "Only a true bride of christ may wield such a holy relic." But in the flashback where it is first used, it's being used by a man, perhaps a Bishop.