Gemara

(Ula citing R. Yochanan): If Kodshim died (without Shechitah), mid'Oraisa Me'ilah no longer applies to them.

Sukah 46a (Rav Sheshes): "Chag ha'Sukos Shiv'as Yomim la'Shem" teaches that Atzei (wood of the) Sukah is forbidden all seven days.

(Beraisa - R. Yehudah ben Beseira): "Chag ha'Sukos Shiv'as Yomim la'Shem" - just like Hash-m's name takes effect on the Korban Chagigah (it is Asur b'Hana'ah), His name takes effect on the Sukah.

Shabbos 45a: R. Shimon forbids Muktzeh for the Mitzvah (alone)!

(Beraisa): One may not take wood from a Sukah (on Yom Tov), only from nearby;

R. Shimon permits.

All agree that on Sukos one may not take from a Sukah for the Mitzvah.

If he stipulated, we follow his Tanai.

Question: How can R. Shimon permit? He destroys an Ohel!

Answer (Rav Nachman bar Yitzchak): He permits to take from a Sukah that fell.

Question: Why does a Tanai help? "Chag ha'Sukos...la'Shem" forbids Atzei Sukah!

Answer (Rav Menasiya): A Tanai helps for a Sukah that is not for the Mitzvah.

Question: A Tanai helps even for a Sukah for the Mitzvah!

Beraisa: If a Sukah had enough Sechach, and one adorned it with sheets and fruits... it is forbidden to use (or eat) them until after Simchas Torah;

If he stipulated, we follow his Tanai.

Answer (Abaye and Rava): In the Beraisa, he stipulated that he may use them at any time during Bein ha'Shemashos (of the first night), so they do not get Kedushah;

i. Atzei Sukah get Kedushah, so they are forbidden all seven days.

Rishonim

Rif (Beitzah 17a, according to the Ran): The Gemara answers that a Tanai not to refrain at any time during Bein ha'Shemashos helps for Atzei Sukah (if the Sukah will fall) as well as for Noyei Sukah.

Rebuttal (Ran Beitzah 17a DH Aval and R. Mano'ach Hilchos Sukah 6:15): Our texts say that a Tanai helps only for a Sukah not for the Mitzvah. Noyei Sukah are different. They are forbidden only because one is Mevatel them to the Sukah. If he stipulates that he may use them, he was not Mevatel them. The Sukah itself is Kodesh due to the verse. Also, the Tanai does not help because one may not dismantle the Sukah during Bein ha'Shemashos due to the Isur of destroying an Ohel on Yom Tov.

Rambam (Hilchos Sukah 6:15): Atzei Sukah are forbidden all eight days of the Chag. One may not benefit from the walls or the Sechach all eight days. This is because they are Muktzeh the entire seventh day, including Bein ha'Shemashos (between the seventh and eighth days). Since they are forbidden all of Bein ha'Shemashos, they are forbidden the entire eighth day.

Rebuttal (Rosh Sukah 1:13): Only the Sechach is forbidden, but not the walls. The Gemara (12a) forbids Atzei Sukah from "Chag ha'Sukos...", which teaches that Sechach must be of chaff of the granary or winepress!

Defense #1 (Korban Nesan'el 60): Perhaps the Rambam expounds the 'Vov' in "Sukos" to include the walls. The Gemara (6b) expounded it to teach the required number of walls.

Rebuttal (Taz OC 638:1): The Gemara (6b) expounded the Vov in a different verse, "ba'Sukos Teshvu..."

Defense #2 (Beis Yosef OC 638 DH v'Yesh): The Rambam forbids the walls mid'Rabanan due to Muktzeh. The Mizrachi says that since they are necessary for the Sukah they are Batel to it, like wood near the Sechach, hence they are forbidden.

Beis Yosef (ibid.): The Rosh challenges the Rambam because he holds that if the walls were not forbidden mid'Oraisa, Chachamim would not forbid them mid'Rabanan.

Objection (Taz 1): The Rosh surely agrees that the walls are forbidden because of Muktzeh. Even Noy on the walls is forbidden because of Muktzeh, all the more so the walls themselves! Rather, the Rambam mentions the Isurim of the walls and of the Sechach together, and the Rosh inferred that he forbids both mid'Oraisa. In truth, the Rambam forbids the walls mid'Rabanan. They are like the Sechach regarding Tanai. (It does not help.)

Question (Rosh, ibid.): Here the Gemara learns the Isur of Sukah from a verse. It is a proper Drashah, for Beis Shamai use the verse for something else and the Gemara asks what is their source for the Isur. In Shabbos (45a and 22a) the Gemara forbids Noyei Sukah due to Muktzeh! (The Ran (ibid.) says that on 22a the Gemara forbids lest the Mitzvah be disgraced.)

Answer #1 (Rosh Sukah 1:13 and Tosfos Beitzah 30b DH Aval, citing the Ri): Atzei Sukah are forbidden mid'Oraisa while the Sukah stands. They are forbidden mid'Rabanan after it falls and the Mitzvah is Batel.

Answer #2 (ibid., citing R. Tam): Atzei Sukah needed to make the Sukah Kosher are forbidden mid'Oraisa. More than this is forbidden mid'Rabanan.

Answer #3 (Ramban, brought in Ran DH u'Mihu): All of these are needed. The verse forbids Atzei Sukah for seven days. Concern for disgrace to the Mitzvah forbids Noyei Sukah for seven days. Muktzeh forbids them even on Shemini Atzeres.

Ran (ibid., DH v'Atzei): The Isur of Atzei Sukah includes the walls, since they are also essential. If one made three walls and later added a fourth, the fourth is not Muktzeh. If one made all four walls at once, they are all Muktzeh.

Chidushei Anshei Shem: Tosfos does not distinguish whether the walls were made at once or not. The verse forbids what is essential for the Sukah. The rest is forbidden mid'Rabanan due to Muktzeh.

Pnei Yehoshua (on Tosfos Sukah 9a DH Minayin): Even though Atzei Sukah are compared to Chagigah, if a Korban died there is no Me'ilah mid'Oraisa. Also, a Korban has Kedushas ha'Guf from when it is made Hekdesh. Therefore, it does not cease. This does not apply to a Sukah.

Question (Pnei Yehoshua Beitzah 30b DH Aval): It is clear from Tosfos in Shabbos (22a DH Sukah) that the answers of R. Meir and the Ri agree with each other. Both are needed. If we distinguish only between what is learned from the verse and what is due to Muktzeh, why do we need to discuss disgrace to the Mitzvah and Muktzeh, which are both mid'Rabanan? In all, there are three reasons! Rather, the verse teaches only about what is needed for the Sukah, while it stands, during the seven days. After it falls it is forbidden only mid'Rabanan due to Muktzeh. What was above the amount needed for the Sukah is forbidden only due to disgrace. If so, what do R. Tam and the Ri argue about?

Answer (Pnei Yehoshua): R. Tam did not want to explain like the Ri, for if so why didn't the Gemara give this simple answer? Why did it ask 'does a Tanai help?' Since it fell, therefore it is permitted! Therefore, he needed to distinguish between the Shi'ur Sukah and above this. This is the Gemara's conclusion; what is above the Shi'ur is like Noyei Sukah. R. Tam explicitly says so in Shabbos. The Gemara initially asked because the Beraisa connotes that a Tanai helps even for the Shi'ur of a Sukah. However, since we discuss a Sukah that fell, why did the Gemara ask 'does a Tanai help?' It seems that R. Tam and the Ri argue about the Halachah. R. Tam holds that if it fell, one may use even the Atzei Sukah during Chol ha'Mo'ed. Therefore, the Gemara could not say that it is Muktzeh for seven days regarding falling. Rather, it is while it is standing, regarding the excess above the Shi'ur Sukah. It is forbidden mid'Rabanan even during Chol ha'Mo'ed, like Noyei Sukah. The Beraisa discusses falling, but that is only regarding Yom Tov. The Gemara mentions the Isur of seven days not to explain the Beraisa, rather, to teach about more than Shi'ur Sukah. A Tanai does not help for this.

Ran (ibid.): Even though Noyei Sukah are forbidden due to disgrace, since they are Batel to the Sukah they are forbidden like it, even if they fall. Therefore the Gemara asked about Atzei Sukah from the law of Noyei Sukah.

Poskim

Shulchan Aruch (OC 638:1): Atzei Sukah are forbidden all eight days of the Chag, both the walls and the Sechach.

Rema: They are forbidden even if the Sukah fell. A Tanai does not help.

Kaf ha'Chayim (13): The Yam Shel Shlomo says that one cannot stipulate not to refrain from it Bein ha'Shemashos, due to the Isur of Stiras Ohel on Yom Tov. If a Sukah was built during Chol ha'Mo'ed, a Tanai to use the excess over Shi'ur Sukah helps for after it falls.

Mishnah Berurah (5): Even according to the Poskim that mid'Oraisa the wood is forbidden only while the Sukah stands, mid'Rabanan one may not benefit from it because it was Muktzeh Bein ha'Shemashos at the start of Sukos.

Sha'ar ha'Tziyon (5): The Yam Shel Shlomo in Beitzah forbids dui even after it falls.

Sha'ar ha'Tziyon (7): The Gemara says that all seven days are like one long day. (Therefore all are forbidden due to Bein ha'Shemashos of the first day.)

Rema: This is only if he sat in the Sukah. Hazmanah (preparation) for the Mitzvah does not forbid.

Taz (3): The Torah equated the Sukah to Chagigah. If so, Hazmanah for Sukah should be significant, just like regarding Korbanos! Why do we distinguish? We do not learn a Hekesh only half-way! Rather, the Gemara did not mean that Sukah is totally like Chagigah. If so, it would have Kedushas ha'Guf! Rather, the Hekesh teaches only that during this time the Kedushah of the Sukah is like Chag. One may not take from it anything that is Mevatel the Kedushah. The Gemara forbids other Hana'os from the Sukah. Surely this is only if it is Mevatel the Kedushah, but one may lean on it or rest something on it, since the Kedushah remains. The Shulchan Aruch said so in 518:8. Here the Beis Yosef brought two opinions in the name of the Ran about whether or not one may smell myrtle (of Sukah walls).