This Halacha is an excerpt from our Sefer
What is one to do with an old and worn out Kosher Tallis?
A Kosher Tallis: A [Kosher] Tallis which is worn out [but still contains Kosher Tzitzis] is to be separated by the person from any use. One is not allowed to clean himself with it, or use it for any belittling activity. [Thus, one may not discard the Tallis in the garbage.] Rather he is to throw it aside and allow it to deteriorate on its own. [This applies for all Talleisim, whether a Tallis Gadol or Katan. This applies even if the Tallis is made of a non-wool material, and is hence only Rabbinically obligated in Tzitzis according to some opinions.]
A non-Kosher Tallis: The above only applies towards a Kosher Tallis, however, a non-kosher Tallis, such as if its strings have torn or the garment is invalid, and may be discarded even in the garbage [after one removes the Tzitzis strings from it, as explained next]. Furthermore, even a Kosher Tallis which is no longer in use and one desires to discard may have its strings removed, hence deeming it invalid, and then be allowed to be discarded in the garbage. See Q&A for the details of this ruling.
What to do with the removed Tzitzis strings: A Tzitzis string that has torn [or been removed] from the Tallis is permitted to be discarded in the garbage bin. Some Poskim however argue and rule that even after the string is torn from the Tallis one may not act with it in a belittling manner, such as to throw it into the garbage or any other repulsive area. Practically, although the final ruling follows the former opinion, nevertheless one who is stringent and meticulous in Mitzvos [to not throw the strings in a repulsive area] is blessed. According to all opinions, the Tzitzis is not required to be placed in Geniza (and may hence be discarded in any area that is not repulsive). However there are some extra meticulous Jews [a Baal Nefesh] who are stringent to place the string in Geniza, or to place the string into a Sefer to use as a bookmark. [However one is not to place the string into a Sefer for Geniza purposes, without intent to use it as a bookmark.]
An old and worn out Tallis that is still Kosher [i.e. with Kosher strings] may not be discarded in the garbage and may not be used for any mundane activity, even if it is not belittling. One is to place it aside and let it deteriorate on its own. A Tallis that had had all of its strings removed, may be discarded even in the garbage. Tzitzis strings that are no longer attached to a Tallis do not require Geniza, and may be discarded wherever one chooses, although one who is stringent should not discard it in a repulsive area. A Baal Nefesh places the Tzitzis strings in Geniza or into a Sefer to use as a bookmark.
What is the status of a Tallis Gadol/Katan that is no longer Kosher but still contains Tzitzis strings?
A Tallis that is no longer Kosher has the same status as a torn string. It may hence be used for mundane purposes that are not belittling, and does not require Geniza, although is not to be discarded in the garbage [due to the strings].
What is the status of a Tallis Gadol/Katan that no longer contains Tzitzis strings?
A Tallis Gadol/Katan that no longer contains any Tzitzis may be discarded into the garbage. [Some write one is to wrap it in a non-transparent bag and then discard it.] However, one may not use it for a belittling purpose such as to make it into pants and the like. One, however, may use it for non-belittling mundane purposes.
If a Kosher Tallis is old and worn out, may its Tzitzis be removed from it so one can discard the Tallis?
Yes. It is permitted to remove the strings even if one does not plan to reuse them, such as if he desires to discard the Tallis and place the Tzitzis strings in Geniza. In such a case, however, one is to untie the strings, or cut off the corners together with the strings, rather than cut the strings off.
 Admur 21:2; Michaber 21:2
 This refers to a Tallis that is worn for the sake of the Mitzvah. [Admur ibid] However if it is a garment that simply contains Tzitzis due to its four corners, then seemingly it may be used for other matters.
 Beis Yehuda 29; Piskeiy Teshuvos 21:1; See B in Q&A regarding the law if the Tallis is no longer Kosher
 Admur ibid; our Nussach in Michaber ibid; However see Kaf Hachaim 21:7
 See 2nd opinion in Admur 21:1 that equates throwing it in the garbage to a belittling activity; Rama 21:1; Orchos Chaim 34 in name of Raavad
 P”M 21 A”A 1; Kaf Hachaim 21:8
 Admur 21:1; Michaber 21:1; Megillah 26b
 Admur ibid; Michaber ibid
The reason: Tzitzis are considered Tashmishei Mitzvah and not Tashmishei Kedusha. This means that they themselves do not contain any holiness and are simply used for the Mitzvah. [ibid; Megillah ibid]
 2nd opinion brought in Admur ibid; Rama 21:1; Orchos Chaim 34 in name of Raavad
 Admur ibid; Rama ibid [In the Rama ibid it is unclear if this statement of “is blessed” is going on those that place the Tzitzis in Geniza, or those that don’t throw it in a belittling area. However in Admur ibid he separates the clauses and learns it to be going on the latter. The practical ramification is whether according to the Rama one must follow the stringent opinion, or only that one who does so is blessed.]
Other opinions: Some Poskim rule that a torn Tzitzis has the same status as a Tallis that is deteriorating which may not be used for a repulsive purpose, or thrown in the garbage. [Birkeiy Yosef 21:1; Levushei Serud on Taz 21:3; Kaf Hachaim 21:2]
 Admur ibid; Rama ibid
The reason: As all agree that the Tzitzis do not contain holiness to require Geniza. [ibid]
 Admur ibid; Rama ibid regarding Geniza; M”A 21:1 and Taz 21:2 in name of Maharil regarding Sefer; Mahariy Viyal 291; M”B 21:8
The reason: As since one has performed one Mitzvah with the string they desire to now perform another Mitzvah with the string. [ibid]
 Makor Chaim 21; Piskeiy Teshuvos 21:1; Mishneh Halachos 11:20
The reason: As one may not use a Sefer as Geniza. [ibid] Likewise, these strings are Muktzah if they are not designated as a bookmark, and hence cause people to move Muktzah on Shabbos.
 Beis Yehuda 29; Piskeiy Teshuvos 21:1 footnote 21
 M”B 21:9
 According to Admur ibid, from the letter of the law it may be used even for a belittling action and may be thrown even into the garbage, although one who is stringent is blessed
 M”B 21:13; See Kaf Hachaim 15:12 “One may remove Tzitzis from an old Tallis in order to discard the Tallis and place the Tzitzis in Geniza”; Piskeiy Teshuvos 21:1
 Piskeiy Teshuvos ibid
 P”M 21 A”A 1; M”B 21:9; Kaf Hachaim 21:11; Piskeiy Teshuvos 21:1; However see Shulchan Hatahor 21:2 that allows using it for even belittling activity.
 M”B 15:3 in name of Artzos Hachaim; Kaf Hachaim 15:12; Piskeiy Teshuvos 15:1