Factual error: In the beginning of the film, the Dowager Empress Marie states that the year is 1916 and that they are celebrating the 300th anniversary of their family's rule. The 300th anniversary actually took place in 1913. (00:01:05)
Factual error: At the beginning of the film, we are told it is 1916, and the Russian Revolutionaries attack. However, the Russian Revolution didn't start until 1917 - on both the English and Russian calendars.
Other mistake: In the song "A Rumor in St. Petersburg," according to the closed captioning, the characters sing the words "no proof of fact." According to the lyrics in the soundtrack booklet, they are actually singing "or through a crack." And if you listen, it's actually what they are saying.
Factual error: Anya receives a sign from the dog to go to St. Petersburg when she leaves the Orphanage in the mid 1920's. St. Petersburg was renamed to Petrograd in 1914 at the start of WWI and again to Leningrad in 1924 following the death of Lenin. The sign should have said "Leningrad". Aside from signs, the whole movie mentions "St. Petersburg" multiple times. There's even a song about St. Petersburg. Even if every old sign could not be changed, there is no reason the people would not use the city's new name. The Bolcheviks would not have looked at them kindly if they didn't.
Factual error: The Czar was not killed until the civil war over a year later, although the film implies he was killed on that November night.
Factual error: Anya reaches St. Petersburg in the mid-1920's, and the man in train station ticket office has the Soviet Crest on his hat. This crest wasn't used until the 1930's.
Continuity mistake: When the Dowager Empress is walking towards the door and says the cinnamon smell is from an oil, her reflection in the mirror exactly matches her movements, including when she turns, when it should show a mirror image. Note the blue sash is on the same side in both. A quick reflection of Anastasia's head just before when she says "I just want to know who I am" has the same problem. After they sit the reflections appears to be right. (01:09:40)