Here is what I could find about the Buturlin clan in my library:
Oddly, the books seldom give first names for any of the Buturlin's they mention, either using initials or just role titles.
The first mention I could find was of an A. V. Buturlin who was a general during the reign of Alexis in the mid 1650's. (Warfare, State and Society on the Black Sea Steppe, 1500-1700, Davies)
The next mention is of a General and Vice-Admiral Buturlin of "the Guards", who was among the nobility that resisted Menshikov's grab for influence during the ascension of Peter II. (Peter the Great, Massie)
The next mention is again of a General Buturlin who was insulted at dinner table by the future Peter III late in Elizabeth's reign. As this would have been some 25-30 years later than above, this may not be the same Buturlin. This General Buturlin was also recounted by Catherine II as a heavy drinker. (Catherine the Great: Portrait of a Woman, Massie)
The next mention is of a Count D. Buturlin who headed a supreme secret committee on censorship as part of Nicholas I's panicked reaction to the revolutions of 1848 in western Europe. This supreme committee -- popularly dubbed at the time as "The Buturlin Committee" -- was called "censorship over the censors", and was the capstone of the extremes of censorship under Nicholas I that reached such heights that the term "forces of nature" was removed from a physics textbook, and the terms "were killed" were changed to "perished" in a history of Roman emperors describing their demises. (Nicholas I and Official Nationality in Russia: 1825-1855, Riasnovsky)
This D. P. Buturlin (we pick up the second initial in this source) who headed the Imperial Public Library was among those who, during a bureaucratic dustup between rival censorship factions, denounced to Nicholas those in the Ministry of Education who had supposedly taken Nicholas' mission of purifying public thought too lightly. (Fighting Words: Imperial Censorship and the Russian Press, 1804-1906, Ruud) Alexander II dismantled the Buturlin Committee during the first year of his reign as part of a sweeping liberalization of censorship in Russia.
I can find no further mention of any Buturlins, even in books focusing on the end of the empire and the revolutions of 1917 such as Orlando Figes' A People's Tragedy and Douglas Smith's Former People: The Final Days of the Russian Aristocracy.
I hope this gives you a trail or two you can follow to find out more about your forebears. Some of them do seem to have had rubbed elbows with emperors and empresses.