Todays Halacha..
Walking in a public domain on Shabbat with a Tallit Pesula a Tallit whose Tzitzit strings are invalid violates the Torah prohibition of carrying in a public domain on Shabbat. (Orach Hayim, Siman 13) Therefore, if a person wearing a Tallit in a public domain discovers that the strings are invalid for use, he must immediately remove the Tallit, and he may not continue walking while wearing the Tallit. This applies even if removing the Tallit will cause him embarrassment, since the consideration for Kevod Ha'beriyot (preserving human dignity) does not override a Torah prohibition.
If, however, this occurred in an area where carrying is forbidden on the level of Rabbinic enactment, such as a "Karmelit," then the person may continue walking with his Tallit if removing it in public will cause him embarrassment. Laws enacted by the Sages do not override the concern for Kevod Ha'beriyot, and therefore one may wear a Tallit Pesula on Shabbat in a "Karmelit" to avoid humiliation.
The Rama (Rabbi Moshe Isserles, Cracow, 1525-1572, author of glosses to the Shulchan Aruch) extended this principle to allow a person to wear a Tallit Pesula in the synagogue on Shabbat if no proper Tallit is available. If he will feel embarrassed sitting in the synagogue without a Tallit, then he may, according to the Rama, wear a Tallit Pesula to avoid embarrassment. It must be emphasized that this applies only on Shabbat, when there is no possibility of purchasing a new Tallit or tying new Tzitzit strings to his Tallit. During the week, however, a person whose only Tallit is invalid may not wear it in the synagogue to avoid embarrassment, as he has the option of acquiring a proper Tallit.
The Bet Yosef (commentary to the Tur by the Mechaber, author of the Shulchan Aruch), however, strongly objects to this position. He argues that Halacha allows a person to continue wearing a Tallit Pesula in a public domain on Shabbat, but it does not permit one to actually put on a Tallit Pesula to avoid embarrassment. The practice of the Sepharadim, therefore, forbids wearing an invalid Tallit in the synagogue to avoid humiliation. Ashkenazim may wear an invalid Tallit in the synagogue on Shabbat to avoid humiliation, but not on a weekday.
Summary: If a person wearing a Tallit in public on Shabbat discovers that it is invalid for use, he must remove it immediately. If this will cause him embarrassment, then in areas where carrying is forbidden on the level of Rabbinic enactment he may remove the Tallit to avoid embarrassment. According to the practice of the Ashkenazim, a person who has no access to a proper Tallit on Shabbat, and will suffer humiliation by sitting in the synagogue without a Tallit, may wear a Tallit Pesula. This does not apply during the week. According to the practice of the Sepharadim, even on Shabbat one may not wear a Tallit Pesula to avoid embarrassment.
