Continuity mistake: The city railroad scenes were filmed in Spain, and many of the forest railroad scenes were filmed in Finland; in both of those countries, most of the railroad track is broad gauge (the rails are more than 5 feet apart). The plains and mountains railroad scenes were largely shot in Canada, where the rails are "standard gauge": 4' 8 & 1/2" apart. The rails can be seen to jump between far apart and closer together more than once as the movie progresses.
Audio problem: During the "peaceful protest" scene, in a close-up of the crowd the mouth movements don't match the soundtrack.
Visible crew/equipment: When Doctor Zhivago comes home after that trolley ride in which he and Lara pass each other without notice, he enters a glass vestibule in the house. As he opens the glass door, you'll see the reflection of a man's face in it. The reflection bears a close resemblance to David Lean, the film's director.
Factual error: The peaceful protesters bear a banner saying "Brotherhood", which is spelt "Братство" in Russian. On the banner a "т" is missing, and the word is written "Браство."