As the traditional axiom goes, the Pope’s power is ad aedificationem non ad destructionem ecclesiae. Therefore, as the opportunity arises and based on our abilities and station in life, we support him in those things that are ad aedificationem ecclesiae and not those things that are ad destructionem ecclesiae.
More precisely, the office of the Pope exists to serve the following ends: the freedom, peace, and exaltation of the Church; the extirpation of schism and heresy (aka the unity of Christians, to put it more ecumenically); and the propagation of the Catholic faith.
Along those lines, what the Pope can do and should do are two different things. Confusing the “can” and “should” is one of the things that causes the separated brethren to see Catholic dogma on the the primacy as different from ancient praxis.
Catholic dogma on the Pope’s jurisdiction focuses on the abstract principles, because the “should” of every situation cannot be foreseen. But there are plenty of times, going all the way back, when many in the Church, including saints, did not support the Pope in some act, because in good conscience they believed it was contrary to those ends of the papacy and therefore he shouldn’t do it (one of the first of such instances–the Easter controversy involving St. Victor I and opposition from St. Irenaeus and many others–is a great example).
Obviously, just like we should with anything else, if the opportunity arises, we should also support him in his personal virtue and not support him in any sins.