While faithful dissent refers to the fallible teachings of the Magisterium (the spiritual authority of the Church), faithful disobedience refers to the fallible temporal authority of the Church. Again, this most often applies to a local Bishop whose orders, rules, or decisions in a particular case are fallible and incorrect. Occasionally, the Pope or the Holy See may make a temporal decision which is erroneous. Even Ecumenical Councils are not infallible when making temporal decisions.
For example, the condemnation of Pope Honorius I, long after his death, by the Third Council of Constantinople. This was an error because the Pope had not taught in favor of the heretical doctrine (that Christ had one will, not two); he merely left the question unsettled, even in his own mind. Note that, during the time of Honorius I, several prominent Bishops held to and taught this heretical view. I disagree with the decision of that Ecumenical Council against Pope Honorius I and so I will not abide by it. The Pope was not a heretic, because the question was as yet undecided by the Church. Furthermore, my theology concerning the Magisterium holds, in contradiction to the temporal decision of that Council, that no Pope can ever fall into the sin of heresy.
In another example, Pope John Paul II and then Cardinal Ratzinger, said that the teaching of the Pope in Ordinatio Sacerdotalis, against the ordination of women to the priesthood, was an exercise of the non-infallible Magisterium witnessing to an infallible teaching of Tradition and Scripture. I disagree; in my view it is a clear example of papal infallibility. I treat it as such and I teach in my writings that it is so, in contradiction to the statements of Pope John Paul II and then Cardinal Ratzinger (who has not revisited the question since his election, to my knowledge).
In a common example, the Bishop of Mostar has been entrusted by the Holy See with decisions about claimed private revelations in his diocese, including Medjugorje. His decision has been clearly and consistently against a supernatural and heavenly origin for the apparitions and messages of Medjugorje. Yet I and many others still adhere to Medjugorje as a source of true private revelation. Many Bishops permit their parishes to send pilgrimages to Medjugorje; many priests and religious have made pilgrimages there. All this in contradiction to the temporal decision of proper Church authority in this matter. But the Bishop of Mostar made a fallible and incorrect decision. And the faithful have the right to disobey an order, decision, or ruling from a Bishop which contradicts the will of God. The Almighty continues to grant these apparitions and messages through His lowly handmaid Mary, so the faithful can answer the call of God. (Note, however, that some Franciscans have been disobedient to the Bishop, refusing to hand over control of certain parishes to the proper authority of the diocese. Such disobedience is not justified by the fact that the Franciscans support the apparitions and the Bishop does not.)
Obedience to the fallible temporal authority of the Church is limited and contingent, because there exists the possibility of errors contrary to the will of God. Obedience to God is absolute and unlimited. Obedience to the temporal authority of the Church is not.
[Quoted from the article: The Limits of the Magisterium]