Here is an example of a civic duty that isn't required by law: Be reasonably well informed as to what the law commands in your circumstances. In particular, someone who goes out of her way to avoid learning what the law requires of her in some circumstance is going against one of her civic duties even if in fact she does not go against the law. There is no general law requiring that one be reasonably well informed about the law, but the role obligations of being subject to the law include a duty to be well-informed.
This suggests an argument against divine command theory. It is one's moral duty to be well-informed about what God commands us. And this moral duty would be in place even if God in fact did not command us anything.
Here's another example. It is one's duty to do what one believes God to have commanded us, at least when doing so does not conflict with what God has in fact commanded us. Thus, if one believes that one has been commanded by God to refrain from eating beef, it is one's duty to abstain from beef even if God did not command it. Now, a divine command theorist might say that in fact God additionally commanded us to do what we think he has commanded us. But it is intuitive that even if God had not commanded us that, we would still be doing something morally wrong if we went against what we think are God's commands (at least assuming that God did not command us to act as we did).
Similarly, one is a bad citizen—one violates the duties incumbent on one as citizen—when one disobeys something that one incorrectly believes to be a just law.