Can we throw away a used Tefillin bag and a Mezuza case, or do they have to go with Shaimos?
The bag that actually holds the Tefillin and a Mezuzah case are both considered a תשמיש קדושה (usage of holiness) and they should be placed in Geniza.
See here regarding the outer plastic “shoulder bag”:
Does a Tallis & Tefillin shoulder bag need to go in Shaimos?
Sources:
See Shulchan Aruch Harav (Orach Chayim 42:6). Shulchan Aruch Orach Chayim 154:3; Mishna Berurah 154:7.
#5287 (2)
#11869