When Ivan the Terrible is talking to Zinaida nearby the time machine the position of vessels and the color of the fluid inside them changes between the shots. Also the water in the fish tank in the background the unclear and muddy in one shot and crystal clear in one shot in the next.
The tape recorder seen besides Shurik's TV has two reels of transparent plastic in one shot and one red and one transparent reel in the next.
Dyak Fyodor is giving the false czar the commanding Ukas and he signs it. However at that time the tradition in the Russian Empire was forbidding ruling people to use feather and ink. The Czair was only putting a Stamp at the end.
The Eggplant caviar which is showed is actually Zucchini caviar. Also eggplant caviar didn't exist in Russia until 17th century.
Dressing up into the czar clothes usually takes a lot of time and requires additional assistance. However Miloslavski and Bunsha gets dressed up very quickly and without further help in the movie.
In the end of the movie Ivan IV Vasilyevich says that he is born in the year 1533. However, Ivan was really born on 25 August 1530. 1533 was the year he became Grand Prince. Furthermore during Ivan the Terrible's time the year was announced after "creation of the world", based on the Konstantinopel period. So Ivan the Terrible should have said "Summer 7038 after creation of the world"
The Czaress Marfa Sobakina died 15 days after the marriage to Iwan The Terrible in the Alexandrov Kremlin cause of a big deceases (apparently she was poisoned). So she wouldn't join a party and wouldn't be so colorful and jolly as described in the movie. Furthermore they were married at the 28 October 1571 and the movie takes place in spring-summer period. However this goof is taken directly from the play "Ivan Vasilievich" by Michail Bulgakov, on which the movie is based on.
When Ivan the Terrible is standing in the elevator, he is crossing himself with three fingers. That wouldn't been done until 1653, 79 years after his death. Before that Russian-Orthodox were crossing themselves with two fingers.
When the characters from XX century Moscow came to the time of Ivan the Terrible they didn't experience any problems in communication. Meantime the language has changed dramatically within 5 centuries. If it's possible that the guests from the future could understand Old Russian, they hardly could speak it without being immediately exposed.
The Czar's orchestra contains a balalaika, which didn't exist during the time of Ivan the Terrible. It was mentioned the first time in the 17th century and it become a standard instrument in the 19th century.
The song "To ne silnaya tucha zatuchilas'" which was sang by the big dinner party was written in 1572, one year after the setting in Ivan the Terrible's period and was dedicated to the destroying of Krim from 30.07-02.08.1572.
The time machine is destroyed by Miloslavski by throwing a halbert. However those only existed in Russia since 1605.
When Bunsha and Miloslavsky are running away from the guards they are running through a white-colored Kremlin. However the Kremlin in Moscow, where the story actually takes place is red, since Ivan III, who came 28 years before Ivan the Terrible. The showed building is the Kremlin in Rostov, 200 km northeast of Moscow.
Timofeef calls the weapon that destroyed his time mashine a bardich. However this is actually a halbert, which is even looked different.
By naming some of the cities Ivan the Terrible has invaded , he also named Revel', the old name of Tallin, Estonia. However historically the Czar was trying to invade the city but never did it.