Both the Russians and the Ottomans at various times claimed to be the successor state to the Roman Empire, but in general that status is something more claimed than natural. The British, French, and Germans all claimed they were carrying on the task of Rome. Mussolini famously considered himself to be the heir of Augustus and a large amount of his rhetoric was focused on restoring Roman virtue to Italian soil. The United States has frequently drawn on a comparison to the Republic, from Washington's Order of the Cincinnati to the National Mall in DC, which is a sort of monumentalized imperial space in many ways reminiscent of the Forum Romanum.

In general, Rome has the status of The Empire in Western culture, and so later empires almost invariably have to deal with it in some way.

After the fall of Byzantium, there were propagandistic sources in Russia that refer to it as the Third Rome. Russian states like Kiev and Muscovy had strong connections to the Orthodox church in Constantinople; Ivan III even married a Byzantine princess, so it makes some sense.

Source: Riasnovsky, A History of Russia, 5th ed.

Another note: this Russian Empire coat of arms apparently has Byzantine characteristics, as well:

http://en.wikipedia.org/wiki/File:Coat_of_Arms_of_Russian_Empire.svg