Hacham David Yosef, in his Halacha Berura (Siman 21; listen to audio recording for precise citation), writes that it is permissible to sleep while wearing a Tallit, and not to mention while wearing a Tallit Katan under one’s clothing. This applies both during the day and during the night. The Tallit differs in this regard from Tefillin, which may not be worn while one sleeps. In fact, according to the Arizal (Rabbi Yishak Luria, 1534-1572) and other scholars of Kabbala, it is proper to sleep with one’s Tallit Katan and to never remove it except for bathing. The Kabbalists teach that wearing a Tallit Katan at all times is effective in neutralizing "Kohot Ha’hisonim" – harmful spiritual forces. This practice is recorded in several sources, including the Birkeh Yosef (Orah Haim 8:7), the Ben Ish Hai (Parashat Bereshit, 9), the Mishna Berura (21:15) and the Kaf Ha’haim (21:15).
There is no need to designate a separate Tallit Katan for sleep; one may sleep with the same Tallit Katan he wears the rest of the day.
The Bet Yosef (work by Maran, author of the Shulhan Aruch) cites the Orhot Haim (Siman 34) as recording a custom not to have one’s Tallit cleaned by a gentile. The Bet Yosef comments regarding this custom, "Never have we heard of or seen anyone being stringent in this regard." In his view, it is entirely permissible to bring one’s Tallit or Tallit Katan to be cleaned by a non-Jew. Although the Mishna Berura (commentary by Rav Yisrael Meir Kagan, 1839-1933) cites those who accept the stringent position, Halacha follows the view of the Bet Yosef that one may have his Tallit cleaned by a gentile. Nowadays, in particular, when cleaning is done by machines, one may certainly bring his Tallit to the dry cleaners, even if it will be cleaned by a non-Jew.
Summary: It is permissible to sleep while wearing a Tallit or Tallit Katan. According to Kabbalistic teaching, one should specifically wear his Tallit Katan while sleeping and at all times, except while bathing. It is permissible to have one’s Tallit cleaned by a non-Jew.
