“So powerful is the need for congruence that when people are forced to look at disconfirming evidence, they will find a way to criticize, distort, or dismiss it so that they can maintain or even strengthen their existing belief.” ~Mistakes Were Made (but not by me)
The authors also cite a study in which neuroscientists found that “reasoning areas in the brain virtually shut down” when people are presented with dissonant information, and emotion circuits of the brain light up when consonance is restored. We do not rest easy until the dissonance is resolved.
“The Lord will never permit me or any other man who stands as President of this Church to lead you astray… If I were to attempt that, the Lord would remove me out of my place, and so He will any other man who attempts to lead the children of men astray from the oracles of God…” 
In my experience, this is a statement and a belief that causes a LOT of dissonance.
We all deal with it in different ways. Some try to distort or dismiss past teachings, or even criticize the teacher (i.e. “he wasn’t speaking as a prophet,” or “it was just his opinion.”)
Obviously there have been contradictions in teachings (feel free to list some if you like, for the sake of the debate). Here is a common one:  Brigham Young taught that Adam was God the Father. Spencer W. Kimball taught, “We denounce that theory and hope that everyone will be cautioned against this and other kinds of false doctrine.” So which of these leaders is leading us astray? If you believe that a prophet cannot teach false doctrine, one of them is definitely leading us astray.
If a doctrine (in my mind) has proven itself to be false, then I can no longer believe it. I cannot accept that “never lead us astray” means “never teach false doctrine.”
This idea is backed up by Bruce R. McConkie. When talking about the ban on blacks receiving the priesthood he said,
“We spoke with a limited understanding and without the light and knowledge that now has come into the world… We have now had added a new flood of intelligence and light on this particular subject, and it erases all the darkness and all the views and all the thoughts of the past.”
Elder McConkie’s statement suggests that sometimes prophets have limited understanding, and even “darkness” in their views (as we ALL do). Notice McConkie did not say “sorry, but the Lord was wrong. We actually did lead you all astray.”
In sharing this with other members, a common reaction I get is “well, if you take the view that a prophet can teach something wrong, then what use is a prophet at all? How could you trust anything he said? Why would you believe any of it? Do you think a prophet is just a guy with some good ideas?”
I resolve this dissonance with the belief that God has given us prophets to guide us in our ultimate goal in life, that of progress and salvation. God will not allow a prophet to do anything that would “separate us from the love of God that is in Christ Jesus our Lord.”
Resolving dissonance is not always bad. In fact, it often leads to pro-social action.  In the case of religious belief, it has spurred me to examine my beliefs and assumptions, discard what I deem to be error, and strengthen my commitment to what I believe is truth.
One final thought: Following a prophet requires a lot of faith, prayer, guidance of the Spirit, and (dare I say) use of one’s mind. May we all have the courage to use our intellect as well as our our feelings, for they are equally vital.
Notes: Woodruff, W. (1890). Excerpts from three addresses by President Wilford Woodruff regarding the manifesto. Sixty-first Semiannual General Conference of the Church.  Young, B. (1852). Adam, our father and our God. Journal of Discourses 1, 46-53.  Tavris, C., & Aronson, E. (2007). Mistakes were made (but not by me): Why we justify foolish beliefs, bad decisions, and hurtful acts. Orlando, FL: Harcourt. From page 18: “Fortunately, dissonance theory also shows us how a person’s generous actions can create a spiral of benevolence and compassion, a “virtuous circle.” When people do a good deed, particularly when they do it on a whim or by chance, they will come to see the beneficiary of their generosity in a warmer light. Their cognition that they went out of their way to do a favor for this person is dissonant with any negative feelings they might have had about him.”