Can we throw away a dried up Esrog, or does it have to go with Shaimos?
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.
See Shulchan Aruch Harav (Orach Chayim 42:6). Shulchan Aruch Orach Chayim 154:3; Mishna Berurah 154:7.