Can we throw away a dried up Esrog, or does it have to go with Shaimos? Same question for a used Tefillin bag.
May be discarded in a respectful manner (such as placing it in a separate bag before putting in the garbage).
Best to use it for another Mitzvah, such as Besamim. Some make out of it jam for Chamisha Asar Bishevat.
It is considered a תשמיש קדושה (usage of holiness) and it should placed in Geniza.
See Shulchan Aruch Harav (Orach Chayim 42:6). Shulchan Aruch Orach Chayim 154:3; Mishna Berurah 154:7.