The Halacha distinguishes between a "Tashmish Kedusha" (accessory of a holy object) and a "Tashmish Misva" (accessory of a Misva). The case of a Sefer Torah and the boxes of Tefilin are examples of a Tashmish Kedusha. When they wear out or are no longer useful they require Geniza (storage in a sacred repository). Even though they themselves are not holy, they serve holy objects. On the other hand, a Tashmish Misva such as the Sechach of the Sukkah may be discarded after the Hag.
The strings of Sisit are considered Tashmish Misva. Therefore, if they become ripped or torn, they may be thrown away, as they have no Kedusha after their Misva is finished. Of course, they should not be disposed of in a degrading way. It is best to leave them in a place where it is known that someone will come and get rid of them. Alternatively, they may be placed in a plastic bag and then placed in the garbage.
The Poskim discuss whether the calendar for Sefirat HaOmer requires Geniza, or, after ripping off the page with the current count, it may be thrown away. Assuming that there are no names of Hashem or other sacred texts, the actual formula for counting, "Today is such and such day" has no intrinsic holiness. Therefore, after the Misva is completed, the page may be thrown away, in a non-degrading manner. This is the consensus of Rav Gidon in his "Yoru Mishpatecha", as well as Rav Nissim Korelitz and Rav Elyashiv.
SUMMARY
Sisit and other Tashmish Misva, such as Sefirat HaOmer calendar, may be disposed of in a respectful manner after they are no longer used for a Misva.
