Due to the "kodesh" (holiness embued by usage) properties of the Lulav used
to perform a Mitzvah it is customary not to bin a Lulav after Succot.
The major custom that I have seen is to store the Lulav till before Pesach
when it is used to burn the Chametz (leaven grain) before this Festival.
Hence the object used to perform one Mitzvah is re-used in the observance
I am always surprised at how long the Lulav does take to burn completely
thus rendering any chametz burned with it as truly not a dogs dinner!!
Another custom that I have seen is that the Lulavim are kept for the Schach
(Succah covering) for future years. Again this uses the object for a
Mitzvah rather than junking it.
Olim BaKodesh VeEin Meridim. We increase holiness around us and do not
decrease it. Hence since the object was used for Kodesh it should continue
Rabbi Adam & Shosh Hill