May One Discard a Torn Tzitzit?

Shalom u’vrachah. Yalkut Yosef, Kitzur Shulchan Aruch, Orach Chaim, Siman 21 - The Law of Invalid Tzitzit Strings.

1. Tzitzit strings, as long as they are attached to the tallit, may not be used for anything else according to halacha, such as tying them for other purposes. However, once the strings have been severed from the tallit, according to the principle of halacha, one can throw them away, but one should not use them for a disgraceful purpose, even though they have been separated from the tallit. And one who is stringent and preserves them will be blessed.

2. Similarly, if a tallit has worn out, it is permissible to discard it, and it is preferable to preserve it. Nevertheless, it is forbidden to use it for disgraceful purposes, such as to wipe oneself with it or to clean the synagogue lights and oil, or the like.

Wishing you success,

Benjamin Shmueli