The Turei Zahav (Chapter 21, Subsection 2) writes that it is worthy not to use the wood of the Sukkah for mundane purposes since these pieces of wood were used for a Mitzvah and it is disrespectful to the Mitzvah were they to be used for other things. Some Acharonim concur and write that there is some lingering sanctity in the wood of the Sukkah, so much so that some pious individuals mark the panels and beams so that they can place them in exactly the same positions the next year, as was the case regarding the beams of the Mishkan.
Maran Rabbeinu Ovadia Yosef zt”l rules that according to the letter of the law, it is permissible to discard of the wood of the Sukkah when one does not need it anymore. This wood certainly does not require Genizah (ritual burial) like other sanctified objects, for the Gemara (Megillah 26b) states that a Shofar, Lulav, and wood from a Sukkah are not sanctified objects and are merely objects used for a Mitzvah and may be disposed of.
Maran Ha’Shulchan Aruch (Chapter 664) writes that one should not step on the Lulav after Sukkot. Maran Ha’Chida writes in his Birkei Yosef that the same applies to the Aravot (willow branches) used on Hoshana Rabba in that they should not be treated disrespectfully. (We have heard from Maran zt”l that the Aravot used on Hoshana Rabba have mystical protective properties.)
Halachically speaking, if one no longer needs the walls or Sechach (roof) of the Sukkah, they may be discarded. Nevertheless, Maran zt”l writes (Chazon Ovadia- Sukkot, page 94) that it is best to leave them in a designated place and not to actually throw them in a trash can, for this is disrespectful. One should likewise not step on the wood of a Sukkah. The same applies to the Lulav in that one should not treat it disrespectfully. Some customarily keep the Lulav and burn it together with one’s Chametz on Erev Pesach so that another Mitzvah can be performed with it.
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