Using Tallis to recite Tachanun:
The custom is to cover the face with a garment upon reciting Tachanun. It does not suffice to cover one’s face using his bare arm. [The custom is to use a Tallis to cover the face during Nefilas Apayim. Those that don’t have a Tallis are to use another garment.]
May one use his Tefillin straps to perform Nefilas Apayim?
There are some who do not wear a Tallis for Davening that are accustomed to use the Tefillin straps of the Shel Rosh to recite Tachanun. Due to various reasons it is improper to do so, and likewise this was never the custom in Lubavitch, as testified by revered Chabad Rabbanim and Mashpim of the previous generations. Practically, Bochurim are not to use their Tefillin straps for Tachanun but rather are to use the sleeve of their jacket or another cloth.
 131/1; M”A 131/2; M”B 131/3; Aruch Hashulchan 131/7; Rama 131/2
 The reason: As one needs to have an interval between his head and the floor upon prostrating on a stone floor, and hence since Nefilas Apayim of today is an extension of the original prostration, we hence cover our faces. [implication of Admur and M”A ibid; See Piskeiy Teshuvos 131/7] Some Poskim however question this reason. [P”M 131 A”A 2] Alternatively, this is in order to avoid placing one’s head on his palm on which is written the sins of a person. [Kneses Hagedola 131/10; Elya Raba 131/2 in name of Manhig; Kaf Hachaim 131/47]
 The reason: As the hand and the face is part of the same body, and a body cannot be used to cover itself. [ibid; 74/3; 91/4]
 The reason: Seemingly this is done in order to merit the Tallis with the performance of another Mitzvah.
 Kneses Hagedola 131/10; Elya Raba 131/2 in name of Manhig; Kaf Hachaim 131/47
 As it only covers a very minute area of one’s face, and does not necessarily prevent one from falling on his palm, as brought above. Furthermore, it is not respectful towards the Tefillin, as ruled in 131/1 that for this reason during Shacharis one is to rest his head on the arm that does not have Tefillin. [See Koveitz Haaros 814]
 Rav Eli Landa inquired on this matter from his father Harav Yaakov Landa OBM, and from the noted Mashpia Rav Chaim Sholom Kesselman and both stated that such a custom never existed in Lubavitch.