Is it permissible to discard a Talit katan with its tzitzit?
If the tzitzis are still attached to the garment it may not be discarded, but we may put them respectfully in a bag somewhere even though it will eventually cause it to get discarded.
If the tzitzis are not attached, the garment may be discarded, however the strings should be treated as above. There is a preference to put the strings in geniza (shaimos).
O:CH 21-1&2, M:B 4, 7, 13, Ginzei Hakodesh 17-13 says in the name of R’ Eliyshiv zt”l, that it is also preferable not to throw the garment of the tzitzis in an disrespectful way.