As a note, the Byzantines were not "successors" of the Roman Empire - they were the Roman Empire. The Byzantines referred to themselves as Roman, and the divide between the two is rather arbitrary.
Yes. Yes, they did.
In 1453, when Mehmet II conquered Constantinople and finished off the Roman/Byzantine Empire for good, he proclaimed himself Kayser-i-Rûm (lit. Caesar of Rome). Moreover, much of the Balkan territory that was directly administered by the Ottoman Empire (as opposed to client states like Wallachia) was referred to as Rumelia. This wasn't the first time the Ottomans used the term, either: Rum crops up in a temporary name for the area around Trebizond (which was renamed from the Rum Eyalet to Sivas Eyalet, possibly to avoid confusion with Rumelia) and the city of Erzurum (which is the rough Turkish pronunciation of the previous Arabic name for it, Arzan ar-Rum; Erzurum has since re-entered Arabic as Ard ar-Rum). And, of course, there's the Seljuk Sultanate of Rum, though one shouldn't conflate the Seljuks and the Ottomans despite the fact that they're both Turkic peoples.
Mehmet II also had every intention of conquering Rome itself. The initial Ottoman forces landed in Italy and captured Otranto, the main army was prepared to follow the next season, and then Mehmet II died. Had this not been the Ottoman Empire, which had and would continue to have a tradition of fratricidal succession, Mehmet II's death probably wouldn't have mattered all that much. But because this is the Ottoman Empire we're talking about, Mehmet II's two chief successors (Bayezid II and Cem) immediately came to blows. By the time it was all said and done, Bayezid packed Cem off to Rhodes and later Rome, where he was "detained" at Bayezid's expense. That was really the best case scenario for the loser in House of Osman's succession.
While Bayezid II and his successors wouldn't claim the title as aggressive as Mehmet II did, it -- like the Ottoman claim to the Caliphate -- would crop up now and again throughout history (like when Suleiman the Magnificent made that fantastic boast about how he "seized the Hungarian crown and gave it to the least of [his] slaves," he prefaces it by reminding the reader that he's Caesar of Rome.
hi! you may be interested in these posts. TL;DR: sort of. There are a few scattered mentions in these posts, amidst broader discussions on the relationship between the Roman, Byzantine, and Ottoman empires
What, if any influence did the Roman Empire have on the Ottoman Empire?
Could one legitimately argue that the Roman Empire fell in 1922 with the collapse of the Ottoman Empire? Why/why not? - featuring /u/tiako and u/Ambarenya
How comes that the Byzantines are considered the successor of the Roman Empire resp. the Roman Empire itself, but the Ottomans aren't? - featuring /u/guckfuchs & /u/talondearg
Why do we say "the Roman Empire definitely fell in 1453" when the Ottoman Sultans claimed to be Roman Emperor, ruled in Constantinople, and kept the Patriarchate? - featuring /u/Tombot3000
Most (all?) of these posts have been archived, so if you have follow-up questions, ask them here & notify the relevant user by mentioning their username