Why did Jesus say, "Do not touch me"?

It would seem that Jesus, newly resurrected, showed himself before the Ascension. Given such an otherworldly endeavour, it may well be that human contact would have defiled him; and as such, was not permitted until after he had been with the Father. This would make sense given the women's penchant for falling at his feet and holding on for dear life. In Mary's case, it would have been her first inclination after having recognized her "Teacher". To read more into it dilutes the practical humanity of the encounter in favour of some theological exercise. It is possible that her exclamation of "teacher," and Jesus' subsequent admonition not to touch him, renders a transformative answer to Jesus' initial question, "who are you looking for"? She may have thus come to the correct understanding of his true nature—and confirms it by exclaiming that she had seen the Lord, not the Teacher, in the following verse.

Jesus had just completed his main task of taking on his person the sins of all the world. He had become God's perfect sacrifice. As a result, he needed to be purified. That could only be completed by his father. Why did this have to be done face to face? The Bible doesn't say. But given the intimate nature of the Father, Son and Spirit relationship, this meeting seems to have been necessary. Until Jesus had been cleansed of this burden, He could not allow anyone to come in contact with him. As fulfilling his role as Most High Priest, He must present the sacrifice, Himself, to God in order for the sacrifice to be consummated. This expressed the nature of temple sacrifices in which the High Priest would burn the sacrifice so it could symbolically be purified and ascend to God. Once done, Jesus could then allow Thomas, for instance, to touch his wounds and also explains why he would not allow Mary to touch him in any manner.

The passage does not make clear how Mary is touching him. H.C.G. Moule speculates that she likely grabbed his arm or hand to try and verify his physical existence. Biblical scholar Raymond Brown has listed a wide array of explanations for his injunction:

• Jesus' wounds were still sore so he did not like being touched
• Kraft proposes that the prohibition was because it was against ritual to touch a dead body
• Chrysostom and Theophylact argue that Jesus was asking that more respect be shown to him. This theory is sometimes linked to the notion that while it was not appropriate for a woman to touch Jesus it was fine for a man like Thomas.
• C. Spicq sees the resurrected Jesus as the equivalent of one of the Jewish high priests who should not be sullied by physical contact
• Kastner, who believes Christ returned in the nude, believes the prohibition was so that Mary would not be tempted by Jesus' body
• Mary should not touch Jesus because she should not need physical proof of the resurrection but should trust in her faith.
• Bultmann sees the phrase as an indirect way of saying that the resurrected Jesus was not at this point tangible.
• According to Moule Jesus' intervention is not a prohibition on being touched, but rather an assurance that the touching is not needed for he had not yet returned to the Father and was still firmly here on Earth. His use of the present tense is said to mean that he should not be touched just at this moment, but could be touched in future.
• Some link it with the next verse stating that they should be read as one to say "don't touch me instead go tell my disciples of the news"
• In John Calvin's commentary he argues that Jesus did not forbid simple touching, but rather that Jesus had no problems until the women began to cling to him as though they were trying to hold him in the corporeal world at which point Jesus told them to let go. Some translations thus use touch for the seemingly permitted actions in Mark and cling for the action Jesus chides Mary for in this verse.
• Barrett mentions the possibility that between this verse and John 20:22 Jesus fully ascends to heaven

There are also a number of scholars who have proposed alternate translations. These are not based on direct linguistic evidence but are rather attempts to synchronize the phrase with other parts of the Bible. There is also some evidence that the wording may have been mangled.

• Some scholars eliminate the negative leaving the phrase as "touch me," implying that Jesus is telling Mary to verify his physical form
• W.E.P. Cotter and others argue that the text should actually read "do not fear me"
• W.D. Morris believes it should read "do not fear to touch me"

Source: John 20:17 (Wikipedia)

Another possibility is that women were considered ritually unclean when menstruating:

Through much of its history, especially in the West, women were considered ritually unclean. According to Jewish tradition, a woman's monthly flow of blood put her regularly into a state of ritual defilement. Similar taboos against menstruation existed in pagan Greek and Roman circles. Through their anti-sex mania, the Fathers of the Church aggravated the fears of women's ritual uncleanness. Church leaders were anxious that such uncleanness might defile the holiness of the church building, the sanctuary and mainly the altar. In a climate that increasingly looked on all aspects of sex and procreation as tainted with sin, theologians considered that an ‘unclean creature’ like a woman could not be entrusted with the care of God's sacred realities. Prohibitions based on the presumed ‘ritual uncleanness’ of women have remained in official Church Law for the last 700 years.

Source: Women were considered Ritually Unclean

The first two references to menstruation can be found in the Book of Genesis. First in describing the patriarch Abraham and the fact that his wife Sarah had reached the age of menopause, "Now Abraham and Sarah were old, advanced in age; it had ceased to be with Sarah after the way of women." The second occurs when Rachel tells her father Laban that she cannot come to meet him since she was in the seclusion that was mandatory for menstruating women: "And [Rachel] said to her father [Laban], 'Let not my lord be angry that I cannot rise before you, for the way of women is upon me.'" Explicit references to menstruation can be found twice in the writings of the prophet Ezekiel. He writes of the ritual impurity that menstruation brings about in a woman and all she comes into contact with during that specified time—"[A man is righteous] if he does not eat upon the mountains or lift up his eyes to the idols of the house of Israel, does not defile his neighbor's wife or approach a woman in her time of impurity." "Son of man, when the house of Israel dwelt in their own land, they defiled it by their ways and their doings; their conduct before me was like the uncleanness of a woman in her impurity." But the most substantial scriptural text, the one that deems women ritually impure for the interval of time surrounding her period is Leviticus 15. This chapter deals with unclean discharges from the genitals in both men and women. The first half speaks of various forms of seminal and venereal emissions and men, and the second of the flow of blood and menses in women. I quote that section, Leviticus 15:19-30 in full: v. 19 When a woman has a discharge of blood which is her regular discharge from her body, she shall be in her impurity for seven days, and whoever touches her shall be unclean until the evening. 20 And everything upon which she lies during her impurity shall be unclean; everything also upon which she sits shall be unclean. 21 And whoever touches her bed shall wash his clothes, and bathe himself in water, and be unclean until the evening. 22 And whoever touches anything upon which she sits shall wash his clothes, and bathe himself in water, and be unclean until the evening; 23 whether it is the bed or anything upon which she sits, when he touches it, he shall be unclean until the evening. 24 And if any man lies with her, and her impurity is on him, he shall be unclean seven days, and every bed on which he lies shall be unclean. 25 If a woman has a discharge of blood for many days, not at the time of her impurity, or if she has a discharge beyond the time of her impurity, all the days of the discharge she shall continue in uncleanness; as in the days of her impurity, she shall be unclean. 26 Every bed on which she lies, all the days of her discharge, shall be to her as the bed of her impurity, and everything on which she sits shall be unclean, as in the uncleanness of her impurity. 27 And whoever touches these things shall be unclean, and shall wash his clothes, and bathe himself in water, and be unclean until the evening. 28 But if she is cleansed of her discharge, she shall count for herself seven days, and after that she shall be clean. 29 And on the eighth day she shall take two turtledoves or two young pigeons, and bring them to the priest, to the door of the tent of meeting. 30 And the priest shall offer one for a sin offering and the other for a burnt offering, and the priest shall make atonement for her before the LORD for her unclean discharge.

Source: University of Dayton