Dominate: Obviously still the Roman Empire... The Dominate is basically the moment the Tetrarchy was applied to the Roman Empire with Diocletian's rule. Besides, since we consider the (Western) Roman Empire to have fallen in 476 AD, why consider it fell nearly two centuries earlier?
San Marino: From what I got on the story behind the existence of San Marino, it would actually be more of a break-away state than a real continuation of the Roman Empire. Marinus was fleeing anti-christian persecutions when he created the abbey that gave birth to San Marino.
Byzantine Empire: It's the Eastern Roman Empire. So yes, it's a continuation of the Roman Empire for me. Given what went on in the Byzantine-specific thread, I don't want to go into detail why I believe such a thing: you'll find my every argument there up until page 11 or 12, after which I stopped posting because I didn't want to repeat myself over and over.
Odoacer's Kindgom: Not really. Odoacer never claimed the imperial title and only claimed to be a servant of the Eastern Roman Emperor. He was however de facto independant and I doubt he solely applied the laws of the Roman Empire.
Ostrogothic Kingdom of Italy: Same as with Odoacer: they didn't claim the imperial mantle and only pretended to be part of the (Eastern) Roman Empire, but they were independant de facto. Plus, Ostrogothic rule in Italy wasn't exactly the same as Roman rule.
Holy Roman Empire: It's only the result of the Pope not agreeing to Irene's coronation as well as part of the process that led to tensions between Rome and Constantinople, the West and the East. The HRE is also far more german in its origin and while it has a bit of the Roman legacy, that heritage is also mixed with the structures of the former germanic realm that formed following the fall of the Western Roman Empire, especially the Franks. In fact, it probably has more in common with the germanic monarchies than with the Roman Empire itself.
Ottoman Empire: Occupying Constantinople and taking over the Eastern Roman Empire's territory doesn't make you a continuation of Rome. I don't think we really count the various Persian dynasty as continuation of the Achaemenid Empire, so why should we not do the same with the Ottoman in regards to Rome? Besides, while the Turks probably borrowed elements from Byzantium like the Germanic barbarians had done with the Western Roman Empire, they also used their own legal system. Turkish rule has very few in common with Roman rule. Furthermore, the Ottoman Sultans themselves never insisted that much on their status as Kaysar-i-Rum: them being Caliph and Sultan was far more important.
Tsardom of Russia: It had valid dynastic claimants until the death of the Rurikid dynasty, even if their claim was arguably very weak. They also shared the same religion as they were both Orthodox Christian. But Russia was never part of the Roman Empire and aside from Cyril and Methodus evangelising Russia, Rome and the Byzantines played a very small role in Russian history. That disqualifies it from being a continuator of the Roman Empire.
Kingdom of Italy: It's the result of the unification of the various states that formed Italy... States that find their origin in feudal Italy, which itself finds its origins in Carolingian Italy. It's clear that these states were influenced by the customs of the germanic invaders that had taken over Italy and while they probably have descent ethnic links to the Romans as well as inherited some of their traditions, it's not enough to say they are a continuation of the Roman Empire.
Venice: The Venezian Republic is more of its own thing than a continuation of the Roman Empire from my POV.
The Papal States: The Pope wasn't the Emperor and I can hardly consider him to be his heir, let alone his continuator. Plus, there is also the fact that the Pope was far more likely to exerce his spiritual power than his temporal one: the Papal States barely expanded between the moments they were formed and the moments they were dissolved.