Question: How do you reconcile the Orthodox practice of clergy and monastics not cutting their hair with 1st Corinthians 11:14, which says: "Does not even nature itself teach you that if a man has long hair, it is a dishonor to him"?
The way this verse is translated by the Revised Standard Version as "Does not nature itself teach you that for a man to wear long hair is degrading to him". The question then is whether this verse is saying it is a dishonor to a man to have long hair, or is this verse saying something about the way a man wears his hair.
It is highly unlikely that St. Paul was objecting to men simply having hair beyond a certain length. In the Ancient world, men having long hair was not regarded as abnormal, but rather was the norm. Most men did cut their hair infrequently. For example, we are told in Scripture of Absalom (the Son of King David who eventual led a revolt against his father): "at the end of every year he cut [his hair] because it was heavy on him—when he cut it, he weighed the hair of his head at two hundred shekels according to the king’s standard" (2 Samuel 14:26). And Absalom's appearance was far from something that was considered to be odd or inappropriate, but on the contrary, we are told: "Now in all Israel there was no one who was praised as much as Absalom for his good looks. From the sole of his foot to the crown of his head there was no blemish in him" (2 Samuel 14:25). Also, in the Old Testament, there were Nazirites, who would take a vow either for a period of time, or in some cases for life, and one of the things that being a Nazirite entailed was that they could not cut their hair while under the vow, unless they came into contact with a dead body (Numbers 6:1-21). Sampson was a Nazirite from birth, and so only had his hair cut once... when Delilah finally got him to explain the source of his strength (which was his Nazirite vow, symbolized by his uncut hair), and then cut off his hair, his strength was gone, and he was taken captive by the Philistines.
Sampson getting his hair cut by Delilah
St. John the Baptist was a Nazirite from his birth, as was made clear by the words of the Archangel Gabriel to the St. Zachariah, "For he will be great in the sight of the Lord, and shall drink neither wine nor strong drink" (Luke 1:15, abstaining from wine being another aspect of the Nazirite vow, cf. Numbers 6:2-3).
It is also fairly clear that St. Paul himself was a Nazirite. In Acts 18:18 we read: "So Paul still remained a good while. Then he took leave of the brethren and sailed for Syria, and Priscilla and Aquila were with him. He had his hair cut off at Cenchrea, for he had taken a vow. And in Acts 21, beginning at verse 15, we read about how St. Paul participated with four other men who had "taken a vow", and made the offerings in the Temple that those who had completed a Nazirite vow would make.
St. James, the Brother of the Lord, who was also the first bishop of Jerusalem was also a lifelong Nazirite, according to St. Epiphanius (Panarion 29.4), which fits the description he is given in Eusebius's History of the Church: "He was holy from his mother’s womb; and he drank no wine nor strong drink, nor did he eat flesh. No razor came upon his head; he did not anoint himself with oil, and he did not use the bath" (Ecclesiastical History 2:23:5).
Not only did Nazirites not cut their hair, but neither did the priests and Levites of the Old Testament "And they shall not shave their heads, nor shall they pluck off their hair" (Ezekiel 44:20 LXX).
The Old Testament practices have not continued in the Church according to the strict letter of the Law of Moses, because for one thing, since the destruction of Jerusalem in 70 A.D., there has been no temple, and thus the sacrificial system of the Old Testament has entirely ceased to be. However, elements of it have survived. And with respect to cutting hair, with the exception of tonsuring, neither monastics nor clergy traditionally cut their hair.
Also, according to the universal iconographic tradition of the Church, Christ Himself, and most other saints are depicted with long hair.
And so, in light of all the evidence for long hair being a part of the Biblical and Ecclesiastical Tradition of the Church, it seems most likely that what St. Paul is referring to in this passage is to wearing one's hair in a particular way -- most likely styled in a manner customary for a woman.
See also:
Concerning the Tradition of Long Hair and Beards, by Archimandrite Luke (Murianka).
Should a pastor have long hair? (an article by a Presbyterian that is of interest).