Leaders sometime indicate that the Prophet (or collective First Presidency and Quorum of the Twelve Apostles) of the Church of Jesus Christ of Latter-day Saints may be infallible (or close to infallible) in some ways.
Infallible when they reveal the word of the Lord
In a letter to Henry Eyring, Joseph Fielding Smith wrote:
There is one place, however, where I feel that men are infallible. That is when they, as prophets, reveal to us the word of the Lord. We have four published works which have been accepted by the members of the Church as standard in doctrine, revelation and government. These are: The Bible, the Book of Mormon, the Doctrine and Covenants and the Pearl of Great Price. We accept [p.144]of course the Bible, as far as correctly translated. It is a well established fact that the copies coming to us based on translations, more or less semi-modern, contain many errors but when the Bible is in full accord with the other records, we accept what is written, whether the things written harmonize with the teachings of science or not.
The most perfect, inspired, unflawed leaders on earth
Sheri Dew, at the Keynote address of the BYU Women’s Conference said:
Some people get tangled up in the question, “but are Prophets, Seer, and Revelators infallible?” I think that’s the wrong question. A better one is “who exactly are Prophets?” They are the ordained holders of Priesthood keys that authorize the Lord’s power to be distributed throughout the Earth. They may not be perfect. They are, after all, still human, but they are the most perfect, inspired, unflawed leaders on Earth. And their only motive is perfectly pure: to help us find our way back home by pointing us to Jesus Christ.
The Lord has promised that if we receive the Prophet’s words as if he spoke them, the gates of Hell will not prevail against us and the Lord will disperse the powers of darkness and cause Heaven to shake for our good. Accepting the gift of a Prophet and following his counsel unleashes a score of promises to help us deal with the turmoil of the last days.