When some of the characters are attending the opera,
"L'incoronazione di Poppea" by Claudio Monteverdi is being performed. It premiered in Venice in 1642, but by the time that the story takes place (ca. 1807), it had been lost and all but forgotten. A score wasn't rediscovered until 1888, and the first modern performance was given in 1905. The anachronism is probably intentional since Monteverdi's tale of the destructiveness of erotic desire foreshadows the events immediately after that scene.
During the battle of Borodino sequence, as the camera pans down the lines of Russian troops sitting in reserve, a soldier can be clearly seen wearing a modern wristwatch.
Towards the end of the Battle of Borodino sequence (Part III - 1812), there are a few seconds where the shadow of the camera can be seen clearly. This is when the camera is moving along a trench, and a cannon falls nearby it.