Gemara

Beraisa: Tashmishei (things which were used for a) Mitzvah may be discarded, such as a Sukah, a Lulav, a Shofar, and Tzitzis;

Genizah is required for Tashmishei Kedushah, e.g. sacks for Seforim, Tefilin, and Mezuzos, cases of a Sefer Torah or Tefilin, and Tefilin straps.

Shabbos 22a - Rav Yehudah: It is forbidden to count coins near (using) the light of Ner Chanukah.

Objection (Shmuel): Does Ner Chanukah have Kedushah (from which one may not benefit)?!

Objection (to the objection - Rav Yosef): Blood does not have Kedushah (yet the Mitzvah to cover it must be done honorably)!

Beraisa: "V'Shofach...v'Chisahu" - one must cover with the same limb he used to slaughter, i.e. his hand. He may not cover with his foot, lest Mitzvos be disgraceful in his eyes.

Here also, one may not count coins near the Ner, lest Mitzvos be disgraceful in his eyes!

Question: May one use Noyei (decorations of a) Sukah during Sukos?

Answer (Rav Yosef): We learn all Mitzvos (that one may not disgrace them) from covering blood.

Rosh Hashanah 33a - Beraisa - Aba Sha'ul: One may put water or wine in a Shofar in order to polish it. One may not use urine because it is dishonorable.

Rishonim

The Rif and Rosh (4:2) bring the Beraisa verbatim.

Hagahos Ashri: Even though we may discard the Aravos taken with the Lulav, one should not step on them.

Rambam (Hilchos Tzitzis 3:9): If the white or Techeiles threads snapped one may throw them in the trash because it is a Mitzvah without intrinsic Kedushah.

Kesef Mishneh: The Ran and Itur similarly explain that Tashmishei Mitzvah may be thrown away only after the Mitzvah has passed.

Poskim

Shulchan Aruch (OC 21:1): If Tzitzis threads snapped one may throw them in the trash because it is a Mitzvah without intrinsic Kedushah. However, as long as the threads are attached one may not use them, e.g. to tie something, for this disgraces the Mitzvah.

Tur: The She'altos (Korach 126) forbids a Yisrael to do his needs with something used to fulfill a Mitzvah, e.g. to tie something with Tzitzis threads, smell Hadasim, or eat an Esrog. We learn from Kisuy ha'Dam. Tashmishei Mitzvah may be thrown away only after the Mitzvah has passed.

Rebuttal (Tur): We learn from Kisuy ha'Dam that Mitzvos may not be done disgracefully. But if a Mitzvah was done honorably, it is not disgraceful to benefit from it afterwards. Tzitzis have no Kedushah; they are Tashmishei Mitzvah, and one may throw them out or enter a Beis ha'Kisei with them. It seems to me that this is even when the Mitzvah applies; it is good to be stringent.

Rejection (of Tur - Beis Yosef DH v'Tamhani): The Tur does not bring reasons to argue with the She'altos. Gemaros explicitly forbid smelling Hadasim or benefiting from a Sukah during Sukos or from Ner Chanukah. If you will say that these are Muktzeh for the Mitzvah, the same applies to Tzitzis! Also the Rambam, Ran and Mahari Avuhav citing the Itur permit Tashmishei Mitzvah only after the Mitzvah has passed.

Defense (of Tur - Bach DH Tzitzis): Tzitzis are a Mitzvah only when one is wearing the garment. It is unlike benefiting from Hadasim or a Sukah, for the time of those Mitzvos is all seven days of Sukos. Rather, we may learn about Tzitzis from Shofar or Sukah after the Mitzvah passed. Even if the Tzitzis are on a garment, one could decide never to wear the garment again, therefore one may benefit from the Tzitzis.

Rejection (Taz 1): The Bach connotes that the She'altos and Tur argue about Tzitzis on a garment that is not being worn now. Surely the She'altos agrees that this is permitted!

Note: Rosh Hashanah 33a forbids polishing a Shofar with urine because of disgrace, even though it is not used at the time.

Defense #2 (of Tur - Taz 1): They argue about benefit from Tzitzis while wearing the garment. The Tur holds like Tosfos (Shabbos 22a DH Sukah), who forbids Noyei Sukah (decorations) even on Chol ha'Mo'ed and even if they fell, due to two reasons. Firstly, they are Muktzeh at the beginning of Sukos (Yom Tov). Secondly, they are Muktzeh to the Mitzvah. The first reason does not apply to Tzitzis. All cases of benefit the Gemara forbids are either when both reasons apply, such as benefit from the four Minim, or when one wants to simultaneously do the Mitzvah and benefit from it, e.g. during the time Ner Chanukah must burn. This disgraces the Mitzvah itself. One can decide not to consider Tzitzis to be a Mitzvah while using them. One cannot do so with Ner Chanukah, for the Mitzvah is to light it to burn continuously for the Shi'ur.

Note: While one is wearing a four-cornered garment, he is obligated to have Tzitzis on it. It is not clear how one may decide not to consider Tzitzis to be a Mitzvah at the time.

Mishnah Berurah (1): All Tashmishei Mitzvah are like Tzitzis.

Kaf ha'Chayim (1): Threads that were detached are permitted, all the more so threads that were not yet attached to a garment, because designation is insignificant. However, since they were spun and twisted l'Shem Tzitzis, one should not act disgracefully with them.

Kaf ha'Chayim (2): The Shulchan Aruch is lenient about Tzitzis of a regular garment. However, the Shulchan Aruch (Sa'if 2) forbids to disgrace a worn out Talis, for it was worn for the Mitzvah, all the more so we do not disgraceful threads of our garments which we wear for the Mitzvah!

Mishnah Berurah (5): A stipulation allows one to use a Sukah. This does not apply to Tzitzis, since one blesses when putting on a Talis, proving that he intends for the Mitzvah.

Beis Yosef (DH Kasuv b'Cholbo): The Kolbo forbids sleeping in a Talis with Tzitzis or giving it to a Nochri to launder, for this is disgraceful. I never saw or heard of anyone who was concerned for this.

Rema: Some say that even after the threads snap one may not conduct disgracefully with them and throw them in a shameful place. However, Genizah is not required.

Source (Darchei Moshe 1): This is from the Kolbo.

Gra (DH v'Yesh Omrim): Maseches Tzitzis says that we use them for shrouds or a saddle.

Mishnah Berurah (7): One may not overtly throw them in a disgraceful place. If they ended up there because they were not put in Genizah, we are not concerned.

Rema (ibid.): Some are meticulous to put them in Genizah. One who is meticulous in Mitzvos will be blessed.

Magen Avraham (1): Mahari Veil says that one should use them for bookmarks in a Sefer. Since a Mitzvah was done with them, he should do another Mitzvah with them.

Kaf ha'Chayim (5): The same applies to everything used for a Mitzvah, such as walls of a Sukah.

Mishnah Berurah (42:11): A Talis bag is not Tashmishei Kedushah. It is only Tashmishei Mitzvah. Therefore, one may use it for Chulin. Even if it holds the Tefilin in their case, this is only Tashmish of Tashmish Kedushah. If one puts in it a Sidur or uncovered Tefilin, when necessary one may use it for Chulin. Since it was made for the Talis, which is Chulin, it is as if he stipulated from the beginning to use it for Chulin.