The Chabad custom of tightening the knot of the Yud after the blessing:
After the blessing, one is to fasten the strap within the knot in order to fulfill the command of “And you shall tie for a sign on your hand.” [Many Chabad Chassidim have a tradition that this means that prior to the blessing one is to open the loop of the knot of the Yud and then fasten it after the blessing. This is done in order to fulfill one’s obligation of tying the knot of the Tefillin daily, as rule some Poskim. The Rebbe, however, as well as other Chabad Chassidim, did not receive this tradition and was hence not accustomed to do so. Likewise, some Poskim completely negate the reason for the practice, saying that it is meaningless and accomplishes nothing in Halacha, thus creating a possible Hefsek between the blessing and the fastening. Practically, the widespread Chabad custom is to make sure that the strap of the knot of the Yud is open prior to the blessing and then to fasten it after the blessing, although those who do not do so certainly have upon whom to rely. The above discussion is only relevant to Chabad Chassidim, as the widespread custom of the world is not to make the knot of the Yud in such a way, and hence its daily fastening is irrelevant.]
Are those who are accustomed to loosening and then tightening the loop of the Yud after saying the blessing, to do so also by the Tefillin of Rabbeinu Tam?
Yes. The purpose of the daily loosening and tightening has nothing to do with the blessing being said [and is only said by Tefillin of Rashi] but rather with the opinion who requires a daily tying of the Tefillin, and hence the above mentioned custom is relevant to both the Tefillin of Rashi and Rabbeinu Tam.
Are those who are accustomed to loosening and then tightening the loop of the Yud after saying the blessing, to do loosen the strap after removing the Tefillin each day, or only the next day directly prior to putting it on?
This matter requires further analysis if there is any difference or preference to one way over the other. One can argue, however, that it should only be loosened right before putting it on and not after removing it in preparation of the next day’s tightening. I am unaware of any tradition in this matter.
 For a full analysis on this subject, see Siddur of Rav Raskin footnote 30 and Miluim 9; Shevach Hayakar p. 27
 Siddur Admur [Letter 7 in Raskin]“After the blessing one is to fasten the strap within the knot in order to fulfill the Mitzvah of: And you shall tie for a sign on your hand [Devarim 6:8].”; See also Admur 25:17 “The blessing over the Tefillin Shel Yad is to be said after placing it onto the biceps and tying it, as the tying of the Tefillin Shel Yad is the beginning of its making” Admur ibid uses the term tying when referring to the Tefillin Shel Yad, as opposed to “Yihadkenu/fastening” when referring to the Tefillin Shel Rosh, hence implying that there is some knot tying occurring during the fastening; This seemingly is done in accordance to the opinion of: Rabbeinu Eliyahu, brought in Tosafos Menachos 35b and Eiruvin 97a; Tikkunei Tefillin 1:187 of the Baal Sefer Haitim, who require a daily tying of the Yud.
The reason: As doing so fulfills the Mitzvah of fastening on the hand itself with this knot, as this fastening onto the hand is being done with the knot. [Siddur Admur] It is unclear as to the intent of this statement, although seemingly it is coming to say that from the verse we learn that the fastening of the Tefillin onto the arm is to be done using the knot itself, and this can only be fulfilled through fastening the knot itself onto the hand. See next footnote for greater depth into the reason.
 Rabbi Leibal Groner [via correspondence] that a certain individual personally was instructed and shown by the Rebbe Rayatz to do so; Shevach Hayakar p. 21 that Rav Zalman Shimon Dworkin stated that so he was taught in Lubavitch and in name of Rav Leibal Groner that so he saw the Rebbe perform [however, in later correspondence he related to me that he cannot ascertain what the Rebbe did]; Sefer Yimei Meleche 3:984 that the Rebbe was told that this is the Chabad tradition by Rav S.Z. Butman and Rav Betzalel Vilshansky, after asking them the intent of the words of Admur in the Siddur; See Ketzos Hashulchan 8:8 that “We have a tradition from Admur to make knot of the Yad in a way that the strap passes through a small loop that is within the knot itself (unlike the rest of the world who does so in a small loop made on the opposite side of the Ma’abarta), and this loop can be opened and closed with the strap that passes through it.” [However, no mention is made by the Ketzos Hashulchan ibid of the concept of doing so at the time of the blessing. See Siddur Raskin ibid] See Taamei Haminhagim p. 624-625 for instructions in how this knot is made; See Ashel Avraham of Butchach who negates the making of the knot in this method;
Explanation: Meaning, that the Chabad tradition makes the knot of the Yud in a way that it contains a loop that can be widened and narrowed by pulling at the strap. Now, this loop is widened prior to the donning of the Tefillin, often being done immediately after the Tefillin is removed, and is then fastened/narrowed immediately after the blessing. This seemingly fulfills the instructions of Admur in the Siddur to fasten the knot onto the arm. [Nevertheless, from the wording of Admur, it is unclear if this is indeed the intent, as perhaps simply fastening the Yud to the hand fulfills this. Nonetheless, this is the tradition received.]
The reason for this daily fastening: A possible reason that Admur requires that the knot itself be tightened daily after the blessing is to fulfill one’s obligation according to those Poskim who rule that one is required to tie the Tefillin daily. [Rabbeinu Eliyahu, brought in Tosafos Menachos 35b and Eiruvin 97a, but negated by Rabbeinu Tam; Tikkunei Tefillin 1:187 of the Baal Sefer Haitim; See Ashel Avraham of Butchach Tinyana 32:52 who records this tradition behind the knot of Admur, although later negates it; See Toras Menachem 1 p. 31 that doing so does not jeopardize the status of the knot, even though it is undone daily; See Siddur Raskin ibid]
 See previous footnote in the reason for the daily fastening
 Rav Eli Landa Shlita relates that his father Rav Yaakov Landa z”l did not do so when putting on Tefillin; Rav Mondshine states that Rav Nasan Gurray asked Rav Shmuel Levitin if the above tradition is correct and he was told that he did not see this custom practiced in Chabad. He likewise states that he asked Rav Yaakov Yosef Raskin, Rav Shlomo Chaim Kesselman, and Rav Rafael Kahn about this tradition of fastening the knot after the blessing, and they replied that they did not see this done in Russia and the first time they heard of this is when they arrived in Eretz Yisrael.
 Testimony of a Yechidus with Rav Itche Meir Gurary, published in Heichal Menachem and Mikdash Melech Otzros Rebbe 4:497, that the Rebbe himself was not accustomed to do so as his father Rav Levi Yitzchak Schneerson did not teach him to do so, and that the Rebbe even stated that doing so possibly constitutes a Hefsek. The Rebbe instructed Rav Itche Meir to seek council with Ziknei Anash as how he should do so. See also Sefer Yimei Melech 3:984 for a record of the above source and other sources and that the Rebbe answered the author that this matters arbitration belongs to Rabbanei Anash; See Siddur Raskin ibid
 See Ashel Avraham of Butchach Tinyana 32:52 who states that this tradition in the name of Admur and its seeming reasoning to fulfill the Mirtzvah accoridng to Rabbeinu Eliyahu is illogical, as a mere fastening of a strap does not constitute making a knot on Shabbos, and hence does not fulfill the obligation of Rabbeinu Eliyahu. The Rebbe ibid stated accordingly that doing so would constitute an interval between the blessing and fastening. See also article of Rav Mondshine in Koveitz Yagdil Torah Yerushalayim 5:81 who questions the above-mentioned tradition and its usefulness; See however Siddur Raskin ibid who defends this tradition of Admur and explains that indeed the closing of the strap is considered like one is making and tying the Yud, and thus fulfills the Mitzvah accoridng to Rabbeinu Eliyahu.
 As perhaps by leaving the loop open it is considered to ruin the knot/Yud and demote it from its status of holiness, and therefore it should only be done momentarily before putting on the Tefillin. [See Minchas Shlomo 2:2 who negates this worry] Furthermore, perhaps if one leaves the loop open constantly and only closes it while he is wearing it enters the knot of the Yud into question how’s the weather this considered a permanent and Kosher knot. [See Avnei Nezer O.C. 183:5, although his ruling is negated by the Poskim and practical custom as brought in Halacha ??] On the other hand one can argue and say that in truth opening and closing the loop does not affect at all the Yud of the knot, which is shaped only on the upper part of the knot which is not touched by the loop and therefore both of the above concerns are irrelevant. Nonetheless, see the wording in Tosafus Eiruvin 97a in negation of the opinion of Rabbeinu Eliyahu that “One is not required daily to open the knot of the Tefillin and then close it at the time of putting it on”; See also Toras Menachem 1 p. 31 that according to Rabbeinu Eliyahu “one is to open and close the knot daily” hence implying that it is be opened that day and then closed, and not open the day before. Vetzaruch Iyun!