Is it possible for the House of Savoy to rule Greece and is it also possible for the Savoy ruled Greece to consider itself as a successor state of the Eastern Roman Empire.
The OTL greek state considers itself a successor state of the Byzantines. It's not a core part of their identity but there certainly is a large acknowledgement that the Empire basically was medieval Greece and played a huge part in their modern culture. Its part of why Istanbul and Izmir are the Greek recidivist holy grail and the Megali Idea was such a huge part of Greek foreign policy despite it being a terrible idea. Them ending up with a savoy prince should also not be too much of a problem. As long as there is a viable prince and the French Faction is dominant.