It's an interesting and challenging passage, as are parallel commands in 1st Peter and Titus. Note that both Peter and Paul tell us what government's responsibility is directly in explaining why we should obey. The question is begged, "Do we obey outside of these examples of state responsibility?"
The answer is not easy, as there is so much variance. I think (I hate saying that on doctrinal matters, but this one is not as cut n dry as many others) that the matter does get right down to the heart of the individual and the situation. Some would accuse me of situational ethics here, which I strive to avoid as much as possible. I suppose it could be considered that, but I really don't think so.
We are to protect our testimonies. To whatever degree we are able, we are to be at peace with all men. If we are to get in trouble with authority, it should be for obeying God.
Some ordinances are merely extortive in nature. They serve no purpose other than to provide income for the state. We might break those without sinning, recognizing that we may have to pay a fee for having broken the ordinance. Paying the fee keeps us at peace.
There's also the fact that nothing in Rome compared to the overreach of today. Yes, their laws could be absolute and oppressive. But they could not record all your transactions, track your passport, etc. So, while I do consider the commands of God to be timeless, I also think that context must be considered so we can make a parallel application in a world that is in many ways much different than the first century church.
In the US the church has become too cozy with the state. Christians have been short-sighted in using the state to push a sort of social gospel through legislation. It's a form of abdication though, and is severely backfiring today, IMO. If the church would have focused on the gospel and not used the state to push morality, I think we'd find it far healthier and more powerful in really helping to change lives for the better, through Christ, rather than dealing with all this backlash.