Miloslavski's quote "Gentlemen, keep your money in your bank, if you have any" is a pun to the commercial slogan of the soviet bank "Bank of the working, saving and loaning people of the USSR", also know as the "Savingbank of the USSR".
In the banquet scene, they display a spoonful of eggplant caviar ("a foreign product") next to bowls of red and black caviar. This is a dig at the lack of consumer goods in the Soviet Union, where red and black caviar were almost impossible to find so Soviet citizens made do with an eggplant confection described as "caviar."
In the scene where Bunsha meets with the Swedish ambassador, the first thing he says is "Hitler kaput!" It's one of the most recognizable phrases in German for many Russians. However, the original script had him say "Peace - friendship!", but the Soviet censors thought it inappropriate. The resulting change ended up being more humorous.
The painting Ivan the Terrible is looking at in Shurik's flat is the painting "Ivan the Terrible killing his son" by Ilya Repin from 1885, describing an event that happened "on Friday, 16 November 1581". Ivan Ivanovich of Russia, Ivan the Terrible's son died actually on 19th November 1581.
Final appearance of Shurik, a character written by Leonid Gaidai.