What do I do with a tallit, when it is damaged and can no longer be used?
That is a great question. The tallit is actually made up of two parts – the garment and the fringes attached to the garment. The fringes, or tzitzit, are the part of the tallit that are used for fulfilling the mitzvah of wearing tzitzit, and should therefore be treated with a greater measure of care. The garment itself, once the tzitzit are removed, may be disposed of. It is appropriate to wrap the garment in paper or plastic before disposing.
The tzitzit should be set aside for burial. Until then, they may be kept in a repository for sacred objects, known as a genizah. You may wish to check with your congregation to see if it has a genizah. If not, perhaps other nearby congregations have one. Check with your rabbi about the proper burial of the fringes. This may be done in conjunction with a Jewish cemetery.