Perhaps the most famous verse on this topic comes from Proverbs 3:5 which says, "Trust in the LORD with all thine heart; and lean not unto thine own understanding." This is probably the first clear scripture verse which indicates that our own reasoning is somehow faulty. The Old Testament is full of these kinds of verses. The New Testament also includes several--such as Romans 8:1 and Philip 3:3. If you move into LDS (Mormon) scripture, it is just as evident that human reasoning shouldn't be trusted. (See the comments made and the scriptures quoted by Bishop Edgley in 1993 that were reprinted in the February 10, 1996 issue of the LDS Church News.)
The most interesting thing about this line of thought is the people who propagate it are the ones who tell you that since your own human reasoning can't be relied upon, you must rely on them or those in authority over them. It's only rarely that they ask you to examine their words for correctness. They claim that since they are God's own servants or because they follow those who they perceive to be such, they are not the 'arm of flesh' or followers of the arm of flesh, but rather they are the ones who know and follow the will of God.
The Lord is supposedly speaking to the church members through Joseph Smith when in D&C 1:38 he states, ". . . by mine own voice or by the voice of my servants, it is the same". Using this reasoning, as leaders of most any church or congregation frequently do, what the leaders are actually saying is not 'trust not in the arm of flesh', but rather, 'don't trust your own reasoning, trust mine instead' or 'your arm of flesh is not to be trusted, but you can trust in my arm of flesh'.
This teaching that solid reasoning is inferior to tradition or a certain leader's opinions is perhaps the biggest error a church can make if it is honestly seeking after the truth (which is what the religions all claim). This backwards version of saying don't trust reason has led to some seriously damaging religious doctrines. For the LDS church, it resulted in racism which couldn't leave the church until God decided to change his mind in the 1970s. It has led to sexism which still hasn't been able to be purged from the church. In fact, it will be incredibly hard to clean the church of its sexist tendencies given what the scriptures say and what men who were thought to be prophets have said.
Now let's move on to the other point of view which has caused some to accuse scientists, objectivists, realists, freethinkers, individualists, and humanists of 'trusting in the arm of flesh'. These people are criticized by religionists for basing their lives on logic, reason, thought, observation, and the scientific method. They are told that if they will only rely on faith, then they can understand reality. Where has that taken humans in the past? By relying on the testimonies of others, religionists have held false assumptions for hundreds and sometimes thousands of years. They thought that the earth was not as we now know it is, that the earth and/or the first humans appeared on the scene only 6,000 years ago (see D&C 77:6 and this site for the LDS view on this), that men lived on the moon, that the Book of Mormon is the 'most correct of any book on earth', and they held countless other superstitious beliefs which are clearly false. Many of these (and other) superstitions are still believed in by religionists.
So who is really relying on the arm of flesh and who is relying on the arm of reality or are they one in the same? Isn't belief in something because it is a tradition or because an authority figure tells you to believe in it relying on the arm of flesh? One of the questions Mormons must answer in the affirmative to be judged 'worthy' to enter the temple is as follows:
"Do you sustain the President of the Church of Jesus Christ of Latter-day Saints as the Prophet, Seer and Revelator and as the only person on earth who possesses and is authorized to exercise all priesthood keys? Do you sustain members of the First Presidency and the Quorum of the Twelve Apostles as prophets, seers and revelators? Do you sustain the other General Authorities and local authorities of the church?"Doesn't this sound more like trusting in the arm of flesh than the scientific method? Is the scientific method really relying on the arm of the flesh when one of its foremost points is to question authority and attempt to prove long held false beliefs to be just that--false?
A book that I highly recommend to anyone who wants to further their study of this topic is Carl Sagan's The Demon-Haunted World: Science as a Candle in the Dark. The chapter entitled 'Antiscience' is especially relevant.