No, Torah does not. In this MLT agrees with the Karaite position and disagree with Orthodox Jews and most Messianics as the following article explains.

In Leviticus 19:27-28 we are commanded:

"(27) You shall not round the edge of your head, nor shall you destroy the edge of your beard. (28) And you shall not make a cutting for the dead in your flesh, nor shall you make a written tattoo upon you; I am YHWH."

In these two verses we are forbidden to make four types of "cuttings":

1) Cutting the head or hair
2) Cutting the face or beard
3) Cutting the flesh
4) Inscribing writing on the flesh

What precisely is forbidden by these four commandments? Are we required to grow long Elvis-style side locks? Or Rabbanite-style "pe'os"? To understand these four commandments we must consider the meaning of the words in their immediate context as well as the broader context of the entire Tanach and the ancient world in which the Torah was given.

Let us begin with the first commandment in the series, rounding the side of one's head. To round the side of the head does not mean to cut the head itself but rather to cut the hair on the head. Specifically we are forbidden from rounding the "Pe'ah" of the head. Pe'ah is often translated as corner or side-lock, but it actually has the meaning of "side" or "edge". This is always the meaning of the word Pe'ah in hundreds of passages throughout the Tanach such as "and for the second side of the tabernacle, on the north side (Pe'ah), twenty boards." (Ex 26:20) and again" And the west side (Pe'ah) shall be the Great Sea, from the border as far as over against the entrance of Hamath. This is the west side (Pe'ah)." (Ezek 47:20).

To "round the edge of your head" means to cut off the hair around the sides of the head. Many exegetes associate this with the pagan "bowl-cut". A bowl-cut was an ancient hair-cut with pagan significance that was created by placing a round bowl on the head and cutting all the exposed hair.

However, when the prohibition to cut one's hair is repeated in Dt 14:1-2 we read: "...you shall not cut yourselves nor shall you place baldness between your eyes, for the dead." Since most people do not have any significant hair "between the eyes" this phrase is usually understood as meaning the hair on the front of the head above the eyes. Bearing this in mind, we learn two things from Dt 14. Firstly, we learn that the prohibition is not necessarily a bowl-cut, but making any baldness around the edges of the head. Secondly, we see that the prohibition is specifically in the context of mourning. That is, one is prohibited to make baldness in the head as an act of mourning "for the dead". In ancient times, when someone died the surviving relatives were so distraught that they cut their skin until they bled and shaved bald spots on their head.

While cutting one's hair may sound like a strange act of mourning to the modern reader, this was a common practice in the ancient world. In fact, the Torah even permits non-Israelites to perform this despised mourning practice in certain contexts. Thus we read regarding the captive Gentile woman: "and she shall shave her head... and she shall cry over her mother and her father for a month of days" (Deuteronomy 21:12-14). As an act of mercy, the Torah allows the heathen women to shave her head while she mourns her recently killed father and mother (cf. Dt 20:13-14).

That making bald spots on the head was a mourning practice is also mentioned by the prophets. Thus we read "And I will turn your feasts into mourning, and all your songs into lamentation; and I will bring up sackcloth upon all loins, and baldness upon every head; and I will make it as the mourning for an only son, and the end thereof as a bitter day." (Amos 8:10) Similarly, we read: "Make yourself bald, and shear yourself for the children of thy delight; enlarge your baldness as the vulture; for they are gone into captivity from thee." (Micah 1:16). These are only two of many verses that relate to the fact that in ancient times making bald spots on the head was an act of mourning along with lamentation, rending of clothes and donning of sackcloth. Thus when we are forbidden in Lev 19 and Dt 14 to "round the side of your head" and "place baldness between your eyes... for the dead" the meaning is that we may not shave our head or any part thereof as an act of mourning or sadness. There is no implication in the commandment in Lev 19 that we must grow side locks or pony tails. The only thing prohibited in Lev 19:27a is to shave the side of the head as an act of mourning. Were one to shave their head for stylistic reasons their would be no prohibition whatsoever.

We have seen thus far that the Israelite is forbidden to make cuts in his flesh and shave parts of his head as acts of mourning "for the dead". In Lev 21 we read of a similar prohibition that specifically applies to the Kohanim (descendants of Aaron). In Lev 21 the Kohanim are forbidden from becoming ritually impure from the dead with the exception of their immediate relatives. After listing the relatives that the Kohen may become impure from, we read:

"(4) A man shall not become impurified by his people to defile him. (5) They shall not make bald a baldness in their head nor shall they shave the edge of their beard and in their flesh they shall not cut a cut." (Lev 21:4-5)

The context of the passage is explicitly defiling oneself for the dead. In this case the Kohanim are forbidden from various mourning practices. Not only are they forbidden from coming in contact with the dead bodies of their deceased friends (vv.1ff.) but they are also forbidden from defiling themselves by making bald spots on their heads, by shaving their beards, and by cutting their skin. We see here that three of the prohibitions found in Lev 19 and Dt 14 are repeated in Lev 21. In all three passages both the implicit and explicit contexts are that of mourning practices. Every ancient person knew that one cut one's skin or shaved one's head as an act of mourning and it was these acts of mourning that are being prohibited in Lev 19. While the mourning connotation of cutting flesh and shaving may not be obvious to the modern reader, we have seen that the Torah itself as well as the later prophets take it as a given that cutting one's flesh and shaving one's head are characteristic acts of mourning along with crying and wearing sackcloth.

It is worth noting that the Nazir makes a vow not to shave his head (Nu 6:5). At the end of the period of abstention, the Nazir shaves his entire head, as we read: "And the Nazirite shall shave his consecrated head at the door of the tent of meeting, and shall take the hair of his consecrated head, and put it on the fire which is under the sacrifice of peace-offerings." The reason the Nazir is permitted to shave his entire head is because he is not doing it as an act of mourning. Similarly, we read in 2Sam 14:26 that Absalom, the son of King David, used to grow his hair long and then shave his head every year. Again, this was not an act of mourning and therefore it was permissible to shave the head.

Given that destroying/ shaving the beard is mentioned in the context of forbidden mourning rites in both Lev 19 and Lev 21, we must ask whether shaving the beard was also a forbidden mourning rite? In other words, is the prohibition to destroy/ shave the beard a general prohibition for all occasions or is it exclusively prohibited as an acts of mourning or sadness.

Perhaps the first clue regarding shaving one's beard is the ritual purification of the Metsora or "leper". We read in Lev 14:9: "And it shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off; and he shall wash his clothes, and he shall bathe his flesh in water, and he shall be clean." We see that in certain contexts a person is required to shave his beard and this is even an act of purification. Similarly, we read about the consecration of the Levites: "And thus shalt thou do unto them, to cleanse them: sprinkle the water of purification upon them, and let them cause a razor to pass over all their flesh, and let them wash their clothes, and cleanse themselves." (Nu 8:7). Again we see that shaving the beard and indeed all the hair is not only permissible but can be an act of purification. In contrast, the prohibition of Lev 19 is to shave the head or beard as an act of mourning!

That shaving the beard was an act of mourning in ancient times is clear from many biblical passages. For example, in the Book of Jeremiah we read about a group of pilgrims mourning the destruction of the Temple: "There came certain men from Shechem, from Shiloh, and from Samaria, eighty men, having their beards shaven and their clothes rent, and having cut themselves, with meal-offerings and frankincense in their hand to bring them to the house of YHWH." (Jeremiah 41:5). We see that these pilgrims were mourning and therefore tore their clothes, cut their skin, and shaved their beards. Clearly then shaving the beard was also an act of mourning along with tearing the clothes and cutting the skin.

The fact that shaving was an act of mourning may shed light on a rather obscure passage that till now has defied explanation. In 2Sam 9:1-4 we read that David sent emissaries to Hanun king of Amon to comfort him over the death of his father. For some reason Hanun became convinced that David's emissaries had not come to comfort him but to spy out the land. In a strange act of retribution he decided to cut off half their beards and send them humiliated back to Israel. Thus we read:

"(2) ...And David's servants came into the land of the children of Amon. (3) But the princes of the children of Amon said unto Hanun their lord: 'Do you think that David does honour your father, that he hath sent comforters to you? has not David sent his servants to thee to search the city, and to spy it out, and to overthrow it?' (4) So Hanun took David's servants, and shaved off the one half of their beards, and cut off their garments in the middle, even to their buttocks, and sent them away."

Up till now it always seemed strange that Hanun and his advisors would suspect David's emissaries of being spies without any seeming justification. Even stranger was his reaction to discovering spies be that he cut off their beards. Bearing in mind that ancient peoples shaved off their beards as an act of mourning "for the dead", it becomes clear why Hanun's advisors doubted that David's comforters had come to pay condolences. Presumably Hanun and his cronies sat in the royal court with torn clothes, cut skin, and shaven beards. When David's men arrived with full beards Hanun's advisors assumed they were not coming to mourn the dead king but to spy out the land. For were they really coming to mourn the king they would have shaven their beards. To teach them respect of the dead and humiliate them at the same time, Hanun ordered that half their beards be cut off!

In summation, Lev 19:27-28, Lev 21:4-5, Dt 14:1-2 prohibit 4 different acts of mourning. These are:

1) Making a bald spot on the head as an act of mourning
2) Shaving the beard as an act of mourning
3) Cutting the skin as an act of mourning
4) Writing on the skin as an act of mourning

Interestingly, the making of tattoos as an act of mourning is the most elusive in the list. It is only mentioned once in Lev 19:28 and then never alluded to again in the Tanach. Reference is made to writing on the flesh as an act of dedication to YHWH (Isa 44:5), but never as an act of mourning. Yet the practice of inscribing the name of the dead loved one in a tattoo still exists to this very day. Recently this practice has come to the attention of the public when it was reported that New York firemen and policemen were inscribing tattoos on their flesh in memory of their deceased comrades.

Courtsey of http://www.karaite-korner.org/shaving.shtml

The Living Bible [says]: "You must not trim off your hair on your temples or clip the edges of your beard, as the heathen do." What is here condemned is not the trimming of the beard, but the shaping of the beard to convey pagan images. It was an ancient custom, for example, to trim the hair of the head and the beard to create a circle around the face - the circle being symbolic of the sun. In this version the mustache was shaved off completely. The wearing of the beard in this case was a means of nonverbal communication - it said, "I am a sun worshiper."

Courtesy of http://www.abcog.org/woman.htm