From pp. 405-6 of Paul Barnett’s “Jesus & the Rise of Early Christianity“:
The writer [of Revelation] speaks of “those of the synagogue of Satan who say that they are Jews but are not” in reference to Smyrna and Philadephia. “Synagogue of Satan” implies that the Jewish synagogue was a source of trouble for Christians living in those cities. There were, of course, communities of Jews living throughout the province of Asia, as they had been for the centuries of the Jewish diaspora (cf. Acts 19:10, 17). References in the book of Acts suggests significant antipathy toward a movement that they would have regarded as schismatical and heretical (Acts 19:33-34; 20:3, 19).
Their hostility may have been increased as a consequence of the disastrous defeat at the hands of the Romans in Palestine (66-70), at which time the temple was destroyed as well as the greater part of Jerusalem. Whereas the former Julio-Claudian dynasty had been generally tolerant toward the Jews, their successors the Flavians were less permissive. The generals Vespasian and Titus, who led the campaign in Palestine, tasted firsthand the fury of Jewish fanaticism. After the humiliation of the Jewish prisoners of war in Rome, Vespasian imposed a tax on the Jews, the fiscus Judaicus, for the upkeep of the temple of Jupiter in Rome. No longer having a temple of their own to provide for in their annual temple tax, they must pay instead for an idolatrous shrine for the hated Gentile king. That Christians were free from this obligation (unless they happened to be Jewish Christians) probably intensified the bitterness.
Yet the fiscus Judaicus did secure a degree of protection from local demands for participation in the worship of “the image of the beast”, that is, the emperor. Christians enjoyed no such immunity; they were exposed to harsh treatment from the authorities. It is possible that the Jewish communities in Smyrna and Philadelphia were called “synagogues of Satan” because their members reported Christians to the local authorities, who would have compelled the Christians to participate in idolatrous ceremonies31.
31 Hemer, Letters to the Seven Churches of Asia, pp. 9-10.