Tsarist Russia was seen as the heir of the Byzantine empire, which was called Basileia ton romaion (kingdom of the romans) by the byzantine. The concept of the third Rome was a dynastic and religious one. The mother of Ivan IV (the first Tsar) was the niece of the last byzantine emperor so they saw themselves as the heir the paialogos dynasty. Moreover Ivan was crowned following (at least to some extent) the byzantine coronation ritual. The most iconic text of the birth of this concept is the letter of Filofey (dated a few year before the reign of Ivan IV) which says: "Two Romes have fallen. The third stands. And there will be no fourth. No one shall replace your Christian Tsardom!"

So they saw themselves as the heir of the byzantines/romans and it influenced imperial titulature and some court ritual but I never heard that it had a significant influence on ruling practices. But it had a huge influence on the diplomatic policy of the russian empire who wanted to "liberate" constantinople from turkish rule (an idea that was an important element in the british intervention in the Crimean war), and protect the all the christian orthodox in general

Don't forget that their were other people claiming to be the true heir of the byzantine. Mehmet II entitled himself "Kayser-i Rum" emperor of the romans after the conquest of Constantinople