May one throw out Hosha’anos?
It is customary [and best] to burn the Hosha’anos with the Chametz.
It is however permissible to throw it out in the garbage in a separate bag.
Sources:
.ראה שו”ע אדה”ז סי’ תמ”ה סעי’ י”ב, סי’ כ”א סעי’ א’, שו”ע ב”י סי’ תרס”ד סעי’ ח’-ט’, ועוד