I found the following paragraph to be an especially helpful way of debunking the idea that Paul, in Romans 13, was telling Christians to obey the government absolutely. The author also hints at just how radical Paul’s notion of “governing authority as servant” was in his context.
Some Christians interpret Romans 13 to suggest that Christians are to give total and unconditional obedience to the state: “Let every person be subject to the governing authorities; for there is no authority except from God, and those authorities that exist have been instituted by God” (Rom 13.3:1). However, a closer understanding of the text suggests our obedience to the state is not unquestioning in nature. A few verses later Paul adds that government is “God’s servant for your good” (Rom 13:4). Thus in Romans 13 Paul recasts the prevailing understanding of the Roman political order by insisting that governments are servants—a fairly remarkable thing to argue at the time of the Roman Empire. Paul thereby relativized the Roman imperial order by refusing to accept the emperor as the ultimate authority and arguing that the emperor was under God—a servant with a particular task to do. Roman emperors (and government leaders more generally) were not gods, not lords, but servants; the emperor was not the only or the final sovereign entity.
Corwin Smidt, “The Principled Pluralist Perspective,” in Church, State and Public Justice. IVP Academic, 2007, p. 143.