The Tsar's son is clearly visible in the closing scene depicting Rasputin's funeral. According to numerous eyewitness accounts though, the Heir to the Throne was absent from the ceremony.
A thick layer of muddy water is shown to cover the bottom of Rasputin's grave. A loud splash is heard when the coffin is dropped into it. In reality, Rasputin was buried in January, in the middle of a cold, Russian, winter, which precluded there being ANY water in the grave. (the Tsar's Diary says 'Minus 12'.)
The Tsarina's Lady in Waiting, Ms Vyrubova the 'Intriguante', is shown to be sauntering haughtily the entire film length, once even pushing the Tsarina's wheelchair. In fact, Vyrubova had been badly crippled in a railway accident a year before, and was herself transported in a wheelchair in 1916, or at best had to use crutches to get about (see 'Memories of the Russian Court' by herself, or other sources).
The episode where Rasputin was lured into a trap by clergymen didn't take place in 1916 but 5 years previously. It didn't go exactly as shown in the picture (no singing 'female lure' has ever been mentioned), though accounts differ as to what actually happened during the encounter.