How can I dispose of old s’chach that is torn and I no longer have use for?

After the Chag, neither the Schach nor the four Minim have any Kedusha. You may use them for any use including burning them. Some people are Noheg to burn the four Minim with the Chametz on Pesach. You shouldn't throw them in the garbage since it is a Bizayon and not appropriate. However you may place them together with pruned branches. Rabbi Yaakov Ariel