I read here about certain Jewish sects prescribing face veil. A related question about the rule was asked here, and i was asked to ask a separate question on Biblical history of the same.
Was such a veil used by the women in Biblical times?
Do any Jewish sects encourage face veil for women today?
If you wanted to get creative with the literature, then the answer is yes. When Yehudah encounters his daughter-in-law, Tamar, at the crossroads, he mistakes her for a prostitute since she had covered her face (ויחשבה לזונה כי כסתה פניה; Genesis 38:15).
The obvious question, of course, is do prostitutes cover their faces? Covering one's face would apparantly be a sign of modesty - not of lewdness. As a result, the Talmud records a rather clever midrash (Sotah 10b), which is quoted by Rashi on our verse. The Talmudic passage is as follows:
א"ר אלעזר שכסתה פניה בבית חמיה דא"ר שמואל בר נחמני א"ר יונתן כל כלה שהיא צנועה בבית חמיה זוכה ויוצאין ממנה מלכים ונביאים מנלן
Rabbi Elazar said, "She covered her face in the house of her father-in-law, as Rabbi Shmuel bar Nachmani said in the name of Rabbi Yonatan: every bride who is modest in the house of her father-in-law will merit to have kings and prophets descend from her."
In other words, when the Torah tells us that Yehudah didn't recognise Tamar because she had covered her face, it means that she had always covered her face when she was in his house. Now that she was at the crossroads, with her face uncovered like that of a prostitute, he didn't recognise her. Stands to reason then that covering one's face with a veil is, in the opinion of Rabbi Shmuel bar Nachmani, a sign of virtuous modesty.
In the Song of Songs (4:9):
לִבַּבְתִּנִי, אֲחֹתִי כַלָּה; לִבַּבְתִּנִי בְּאַחַת מֵעֵינַיִךְ
Thou hast ravished my heart, my sister, my bride; thou hast ravished my heart with one of thine eyes
The Torah Temimah, based on a statement in the Talmud Yerushalmi comments:
לפנים בעת שהיו נוהגות הנשים ללכת עטופות היו מגלות רק עין אחת כדי לראות מהלכן, ומכאן רמז שמנהג כזה הוא מנהג כשר וצנוע, שהרי כן משבחה הכתוב שלבבתו בעין אחת
In the former times, when women were accustomed to going out "wrappped up", they would reveal just one eye so they could see their path. And from here is implied that this custom is a proper and modest custom, for the Scripture praises her that she ravished his heart with the one eye.
[A mishna in Shabbos (6:6) states that Jewish Arab women used to go out "shawled" ("רעולות"). Mishna 8:3 notes that the minumum amount of eye-makeup that would be considered carrying on the Shabbos is the amount for one eye. As the gemara (80a) says, women used to only apply makeup to one eye. Rashi there understands that they used to go out wrapped up (מעוטפות), and would only have one eye revealed. All this is not really referring to Biblical times, however the Yerushalmi that the above Torah Temimah is based on (or at least his version of the Yerushalmi) quotes the above pasuk as support for this notion. (H/t @ShimonbM.)]
With regard to those doing so today, and their endorsement and condemnation of various rabbis past and present, please see the Seforim blog post linked to in my answer to your other related question.
There is a fair amount of evidence that, in Biblical times, veils were symbols not of modesty, but of its opposite! To wear a veil was to actively increase one's beauty, or perhaps even to signal one's availability for sex.
(This idea is mentioned in some comments, but I'm here just putting it into a full-blown Answer.)
In the story of Tamar playing the harlot (Gen 38), putting on a veil is one means by which she disguises herself as a prostitue (v. 14) and, indeed, Judah recognizes her as a prostitue because of the veil (v. 15).
The veil is also used as a means of enticement/attractiveness/sexuality when Rebecca is being led by Abraham's servant to meet for the first time her new fiance, Isaac. (Gen 24) Upon being told that the man in the distance is in fact Isaac, she puts her veil on. (v. 65) Mind you, she had no veil on for the entire journey with Abraham's servant - APPARENTLY, there was no "modesty requirement" compelling her to wear a veil when with the servant. Rather, when she meets her fiance - someone who she wants to and should look sexually attractive for! (see v. 67) - she then decides to put on a veil.
(Much of this answer is developed at length by Olivia Wizniter, at http://curiousjew.blogspot.com/2008/02/symbolic-veil.html )
First of all, Jewish sectarianism is a confusing subject to say the least because we draw the lines differently than other religions in several ways. That being said, the sect you link to on wikipedia is more of a schism than a sect. The only verse in tanach I can recall off hand that discusses face veils for women is Genesis 38:15.
