What is the source of the custom to keep the Lulav untill Erev Pesach and burn it with the Chometz?
Kaf Hachaim, 664:60, tells us what to do with the lulav and esrog after Hoshanah Rabbah (the last time we use the lulav), as well as the aravos used for hosha'anas. I'm translating this from the hebrew, so I may have gotten some of the details wrong. Please correct me if I got something wrong:
- After prayer on Hoshana Rabbah take the lulav (with hadasim and aravos still attached) home and put it over the door of the house we sleep in until Pesach, to guard it.
- Then, in the morning before pesach, we take half the lulav and burn it with the chometz.
- Take the other half and burn it in the Matzah baking oven.
- Turn the esrog into jam and eat it on the night of Tu B'shvat, the new year for trees
- give some of the esrog jam to pregnant women, to have an easy labor, and children that will grow up לחיים טובים ולשלום.
- Take the Aravos used on Hoshanah Rabbah for Hosha'anat and put them by the head of your bed, to show how special they are to us. (this will also help protect from fear at night as Sefer Hamidos l'Maran [R' Nachman of Breslev] tells us that Klapped Hosha'anat are a Segula for removing fear).
- If one is really scared at night, put the Aravos under his head while he sleeps.
See here as well (scroll down to Halachos for Erev Shabbos Kodesh, Hoshana Rabbah), for more sources that I didn't look up, including a Talmud Yerushalmi)
Ultimately the source is from the Talmud:
Rabbis Ami and Asi would make a meal of the bread that was used for an eiruv, stating since it was used for one mitzva, let us use it for another (Talmud Shabbos 117b).
Rema (664:9) we put away aravoth and use them to bake matzo, for the reason aforementioned.
Since matzo baking has been commercialized, the custom developed to use it instead for the burning of the chametz.
Similarly, Tur (297) and Sh"A (ibid:4) re: besamim of havdalah, we use the hadassim left over from the lulav, for the same reason. Other examples include using the wax drippings from shul candles for Hanukkah (Bach 673), and using leftover tzitzit as a sefer bookmark (Magen Avraham 21:1).