So the gens Furia did not actually die off, it continued into the Principate, but with greatly diminished prominence. If it had disappeared that would be nothing particularly special, as by the late Republic the plebeian nobility had pretty much taken over the senatorial class completely--only 14 patrician families still had extant patrician lines by Caesar's lifetime, and the vast majority of the prominent politicians of the late Republic were plebeian. The last Furian consulship I know of is in 196, the consulship of Furius Purpureo who failed to win the censorship of 183. Following that holders of the name appear to be mainly equestrian, and it's not totally clear whether these people are really a plebeian branch of the family or whether, like Cicero (who bore the name Tullius but was not part of the line of Servius Tullius), they're just a family with a related name. At the very least the Camillan line apparently continued at least in name, although it's hard to tell whether this is one of the imperial patrician lines revived artificially by Augustus. Claudius' first wife was Livia Medullina, whose father M. Furius Camillus won triumphal insignia against Tacfarinas and who, according to Plutarch, claimed descent from the dictator's line