Hi, long-time lurker. But I think I can actually contribute a bit to this!

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Banned Beards

I actually don't know too much about the Jehovah's Witnesses, so I won't try and touch on their beliefs and practices.

Mormons (LDS), however, I do know a bit about. For the LDS Church and its members, a lack of facial hair is more cultural than spiritual; there's no commandments or doctrine that insist men be clean-shaven. The early Presidents of the LDS Church, from Brigham Young to George Albert Smith (#2 through #8), had varying degrees of facial hair, and many of the other leaders of the LDS Church (early Quorum of the Twelve Apostles and First Presidency) sported what look like quite stylish beards for the time.

Instances within the LDS Church where facial hair is banned do exist. The two that come to mind are (1) young men serving as full-time missionaries and (2) students at Brigham Young University (BYU). From the official Missionary Handbook:

Keep your hair relatively short (not clipped too close) and evenly tapered. Extreme or faddish styles— including spiked, permed, or bleached hair or a shaved head—are not appropriate. Sideburns should reach no lower than the middle of the ear. (See the pictures of a missionary haircut included with your call packet.) Elders should shave each day.

(Missionary Handbook: Missionary Conduct, emphasis added)

BYU's Honor Code contains a similar requirement that "men are expected to be clean-shaven; beards are not acceptable." According to the Salt Lake Tribune, BYU President Ernest Wilkinson banned beards in 1969. (Salt Lake Tribune, The History of BYU's Honor Code," 29. Oct. 2016.)

Both bans are addressed by Dallin H. Oaks, who was President of BYU at the time (he now serves as a member of the LDS Church's highest governing body).

"Our rules against beards and long hair have the same purpose as the requirements our Church makes of its missionaries. In this university, which is largely supported by the tithes of faithful members and which stands as a beacon of Latter-day Saint values, we wish to avoid an appearance that has become associated with rebellion and rejection of values we hold dear. A recent book by Jerry Rubin, the clown prince of the hippy movement, gives this vivid characterization of the meaning of long hair:

"'Long hair is communication. Young kids identify short hair with authority, discipline , , ,—and long hair with letting go, letting your hair down, being free, being open. Wherever we go, our hair tells people where we stand on Vietnam, Wallace, campus disruption, dope. We’re living TV commercials for the revolution.

"'Long hair is the beginning of our liberation from the sexual oppression that underlies this whole military society.'" [Jerry Rubin, Do It, pp. 93, 95–96]

(Dallin H. Oaks, "Be Honest in All Behavior," 30. Jan. 1973, speeches.byu.edu.)

Similarly, in a 1971 issue of the LDS Church's publication for its teenage members, the New Era, Oaks was again cited as saying:

The rule against beards and long hair for men stands on a different footing. [. . .] Unlike modesty, which is an eternal value in the sense of rightness or wrongness in the eyes of God, our rules against beards and long hair are contemporary and pragmatic. They are responsive to conditions and attitudes in our own society at this particular point in time.

There is nothing inherently wrong about long hair or beards, any more than there is anything inherently wrong with possessing an empty liquor bottle. But a person with a beard or an empty liquor bottle is susceptible of being misunderstood. Either of these articles may reduce a person’s effectiveness and promote misunderstanding because of what people may reasonably conclude when they view them in proximity to what these articles stand for in our society today.

In the minds of most people at this time, the beard and long hair are associated with protest, revolution, and rebellion against authority. They are also symbols of the hippie and drug culture. Persons who wear beards or long hair, whether they desire it or not, may identify themselves with or emulate and honor the drug culture or the extreme practices of those who have made slovenly appearance a badge of protest and dissent. In addition, unkemptness—which is often (though not always) associated with beards and long hair—is a mark of indifference toward the best in life.

(Dallin H. Oaks, "Standards of Dress and Grooming," Dec. 1971 New Era, www.lds.org, emphasis added.)

The prohibitions of the LDS Church on facial hair are a product of its members' desire to be "in the world, but not of the world"; a push by their leaders to separate themselves from "worldly things" but remain "the light of the world." And in the late 1960s and 1970s, when counterculture in America was marked a preponderance of beards and facial hair in general, the leaders of the LDS Church wanted their young adult followers to keep a safe distance from counterculture movements that espoused beliefs or practiced activities that did violate LDS commandments, such as drug use or sexual promiscuity. And even though society has moved beyond that, with facial hair rapidly becoming mainstream over the past several years, the unofficial stance of the LDS Church has not drastically changed. (It is worth noting that several publications maintained by the LDS Church have moved towards a more lenient stance, such as the LDS Employment Services' recommendation for "Dressing for Success" suggest that job-seekers "follow appropriate business culture for facial hair.")

TL;DR: Mormons don't have facial hair because of the association it had in the 1960s/1970s with practices they forbid, such as drug use.

I can't say anything about Jehovah's Witnesses or LDS and their facial hair. I can say that all Abrahamic religions, which would include Jews, Christians, Muslims, and (more to the point of your question) Rastafari, have inherited a ban on cutting the beard from the Torah.

Here, it's stated in Leviticus 19:27, two translations:

ESV - You shall not round off the hair on your temples or mar the edges of your beard.

Douay-Rheims (older, Catholic) - Nor shall you cut your hair roundwise: nor shave your beard.

Leviticus 19 in general spells out other rules of purity and decency, like not creating hybrid animals, not wearing clothing made from blended fabrics, not turning your daughter into a prostitute, so this might be a thing that was seen along those lines - beard-trimming apparently has something in common with dishonesty or following false idols (things also banned in that chapter).

There are other statements of this prohibition elsewhere in the Hebrew Bible (which is the book that makes Abrahamic religions Abrahamic), particularly in the context of mourning traditions - the next verse in Leviticus bans cutting the body in memory of the dead (which I believe is where the Jehovah's Witness ban on blood transfusions and tattoos comes from - the idea of penetrating the body with metal - but I couldn't quote anything authoritative on that).

Leviticus 25 further bans the priesthood from trimming beards , which in context appears to be a sign of "profaning" oneself.

Outside the Torah, the idea of cutting beards comes up a few times, generally as a kind of debasement or misery, and as a "foreign" thing... something the neighboring tribes or nations did, that the People of the Covenant weren't supposed to do.

In Jeremiah 41:5 we've got some unhappy foreigners gashing their bodies and trimming their beards after the (Babylonian-appointed) governor of Israel was assassinated... and a couple of chapters later, we've got the Lord saying he's fixing to destroy "those who cut the corners of their hair", meaning the people of Kedar and Hazor, "men of the East" who were defeated by Nebuchadnezzar, the Babylonian King. (This is usually interpreted as the hair on the temples - maybe sideburns. The peot ringlets you might see on Chasidim are specific to these "corners" - Hebrew for "corner" being peah.)

Isaiah has more mourning foreigners, and Second Samuel has some Israelite envoys being abused by shaving. They're told by King David to stay away, in fact, until their beards grow back. How humiliating!

Ezekiel and Isaiah also have references to beard-shaving as an almost apocalyptic thing - something that happens when God's judgement is rendered on Israel. An embarrassment and a calamity, in other words.

From the way it's written about in the Tanakh, I'd assume there are traces of other Ancient Near Eastern traditions about beards and shaving, but it'll have to be up to /u/yodatsracist or /u/captainhaddock or some other, wiser, head to fill in those spaces.