Romania didn't exist at the time of the fall of Constantinople. The states which we could consider Romanian were mostly being occupied of vassalized by the Turks (who themselves claimed to be the heirs of Rome) so any claim to being Third Rome wouldn't have ended well for them

Russia's claim wasn't a result of "Roman-ness" but because of marriage ties with the Byzantine Emperors. The marriage of a Byzantine Princess to a Russian King gave the Russians a claim to the throne (in their minds, anyway) The fact that Constantinople was incredibly strategically important also played a part in this decision

Quite a lot of people claimed the Third Rome title over the years. Bulgaria, Serbia (I believe) Trabizon, and a few other Eastern European countries.

The reason the Western Latins didn't claim this, despite also being Roman in stock, was because for them the Byzantines/ERE weren't really the true successors of Rome. For them, the Roman Empire fell in 476.

Another reason was the HRE. The HRE styled and was at times more and at times less seen as a successor of the Roman Empire in the West. Given that the HRE and it's Emperor sometimes ending up being the most powerful ruler in Europe (Hey, Charles V!) claiming that the HRE's entire existence and raison d'etre was wrong and that therefore you were the real successor to Rome wouldn't have been the smartest move