In Istanbul we hear a muezzin giving the standard Muslim azan (call to prayer) in Arabic: "Allahu Akbar! Allahu Akbar!" However, the movie is set during the 1930s when Mustafa Kemal Atatürk was in power. During this time, the Arabic azan was outlawed, and a Turkish one ("Tanri Uludur!") had to be used instead. After Ataturk's death in 1938 the law was repealed.
The British officer who escorts Poirot on the ferry thanks him for saving the honor of the British garrison in Jordan. The Kingdom of Jordan did not exist until 1946. In 1935 it was the Emirate of Transjordan.
Just after Poirot sips his liqueur (the green drink) in the dining car, a blurry white sedan can be seen zipping across the landscape in a way that cars do not move in the 30s. It appears to be a sedan from the 70s.
As Poirot goes to leave the car after announcing his solution to the murder, Pierre is shown opening the salon door, and holding it open as Poirot pauses in the doorway, turns and watches as the various passengers make toasts to one another. Pierre is the first to step up and raise a toast with his glass of champagne. But when all the toasts have been made, Poirot is shown still standing in the doorway, then turning to exit, even as Pierre (actually just his arm is visible, but it couldn't have been anyone else) is seen in the exact same position, still holding open the door for Poirot.
When the body of the victim, Ratchett, is examined by the doctor and Poirot its eyes are wide open. In fact, the doctor asks permission to close the victim's eyes. However, when the murder scenario is recapitulated by Poirot, as soon as he becomes unconscious from the the sedative administered in his nightcap Ratchett's eyes close.
When Poirot uses the hat box to decode the burnt paper, in one shot Poirot places his small burning lamp to his right in the next shot the lamp is in the center ready for the hat frames to be placed over it.
When Hardman introduces himself to Foscarelli and Beddoes, he tells them to "call me Dick", indicating that his first name is Richard. Yet his name, as Poirot points out during Hardman's interrogation, is Cyrus B. Hardman.
In the train station we see Ataturk's (founder of independent and modern Turkey) poster on the wall. But in the 1930s, Ataturk was alive and had great power. Posters of him were only put on the walls years after his death.
While passengers are boarding the train, you can hear through
the speakers in several different languages where the train is going to stop. Sofia, Belgrade, Zagreb, Brod, Trieste. The city of Brod between Zagreb and Trieste is in Slovenia and not on the railway line. The train halt is at Slavonski Brod located between Belgrade and Zagreb.
The film uses a French Railways (SNCF) Class 230G locomotive, including at the departure from Istanbul Sirkeci station. But locomotives on the Orient Express were provided by each State Railway system, and were usually changed at national borders. Only the carriages, not the locomotives, would have worked right through between France and Turkey.
When the rescue locomotive arrives to push the snow away, (time frame 2:04:17) you can see the snow is flowing smoothly like a liquid, some even floating in the air, it is quite obvious that the snow is really foam from soapy liquid.
The goof items below may give away important plot points.
After the conspirators stab Ratchett, Pierre "cleans up". He puts the blankets back over Ratchett's body, closes Ratchett's door into the passageway outside, locks it, and chains it - all without wearing gloves. While his fingerprints would be expected to be on the doorknob and possibly even the door lock, there would be no reason for his fingerprints to be on the door chain.