The Shulhan Aruch rules that one is not allowed to sell a Tallit with Sisit to a non-Jew, out of concern that he may use it to masquerade as a Jew and ambush an unsuspecting Jew to murder him. Although, the Hayeh Adam (Rav Abraham Danzig of Vilna, 1748-1820) suggested that this does not apply nowadays, the Kaf HaHaim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) maintains that it is still relevant. In fact, Arabs have dressed up in Sisit and Peyot to infiltrate.
The Mekor Haim adds that one should not even sell a Tallit without Sisit to a non-Jew for the same reason. That which the Mishna Berura (Rav Yisrael Meir Kagan, 1839-1933, Siman 20:8) permitted selling a garment after the Sisit have been removed refers to a regular four-cornered garment and not a Tallit designated for prayer.
Rashi brings an additional reason for this prohibition. He says that the Non-Jew may use the Tallit as payment to a Zonah (prostitute), and this may present the impression that it was given by a Jew, causing a Hilul Hahsem.
Nowadays, one may not purchase Sisit without a reliable Hashgacha (seal of supervision). In the times of the Shulhan Aruch it was permitted to trust even a non-Jewish merchant.
The Poskim discuss whether it is permitted to use the Tallit to clean one’s eyeglasses. The Yalkut Yosef and Rav Elyashiv distinguish between a Tallit Katan, which is permitted, and a Tallit Gadol, which is designated for Tefila and may not be used. Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995) ruled that even a Tallit Katan may not be used.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) rules that one may not wipe his dirty hands on neither a Tallit Gadol nor Tallit Katan, whereas he may dry his wet hands on the Tallit Katan, but not the Gadol. For the same reason, the Kaf HaHaim (154:29) prohibits laundering a Tallit with other garments that are heavily soiled. The Poskim do permit using a Tallit for a non-degrading purpose, such as covering something or using it to shade from the sun.
One may not sell a Tallit to a non-Jew.
One must purchase Sisit with a reliable Hashgacha.
One may not use the Tallit for a degrading purpose, such a wiping his soiled hands, but he may use the Tallit Katan to clean his eyeglasses or dry his wet hands.