Sacred objects themselves cannot just be thrown in the garbage; they need to go through a special burial process.

A sacred object is an object on which is written either G‑d’s name, three consecutive words from a Biblical verse, a chapter from the Talmud, or a text of Jewish law or the like. A Torah scroll or the parchment of a mezuzah are examples of sacred items.

A sacred item is an item where G‑d’s name, three consecutive words from a Biblical verse or a chapter of the Talmud,

But what about items that are non-sacred but are used to assist in the performance of G‑d’s commandments, for a mitzvah?

In Jewish law, there are two general categories of objects used for a mitzvah:

  1. Objects that are themselves used to perform a mitzvah, such as: the ram’s horn used as to blow the shofar during the High Holidays and the month preceding them; the sukkah hut we sit in during the holiday of Sukkot; and the tzitzit, the fringes on the tallit prayer shawl.

    We are permitted to dispose of these items; however, it needs to be done respectfully, such as in double wrapping the object in paper or plastic before being putting it in the garbage.

    Note, that during the time that these items are actually in use for the mitzvah, they cannot be used for anything else. For example, it is not permitted to hang something on the tzitzit.1
  2. Objects used as accessories to sacred items, such as: the covering of a Torah scroll; or the case where tefillin, the ritual boxes worn during weekday prayers, are kept when not in use.

    These items need to go through the same burial process as the sacred items themselves.2

Competent scribes are very careful to wrap a mezuzah in parchment paper, or even better, saran wrap, to protect it. The mezuzah is then placed in a case, which is affixed to the doorpost.

The paper or plastic that directly covers the mezuzah is for sure considered sacred and needs to be buried. since mezuzah cases, which are there for beauty3 or to protect the mezuzah, would in most cases also be considered an accessory for a sacred item.4

However, the nails or tape used to put them up can be discarded respectfully.

See Proper Disposal of Holy Objects.