When eventually the boy's parents decide to go to church for help, behind the church a modern 1960s (or later) building is clearly visible in the background, such architecture trends did not exist in the 1950s, yet.
In one of the early scenes, Father Bowden shows a college classroom images of the devil. One of the images was the popular devil head drawn by the artist 'Coop'. The problem is that this movie is set in the 1950s and Coop hadn't even been born by that time.
When Robbie is circling the bed singing "Ave Maria" his mouth in no way matches the words heard.