Gemara

Rav Sheshes: "Chag ha'Sukos Shiv'as Yomim la'Shem" teaches that 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 take effect on the Korban Chagigah (it is Asur b'Hana'ah), His name takes effect on the Sukah.

Beitzah 30b - Mishnah: One may not take wood from a Sukah, only from what is near it.

Question: One may not take from the Sukah (i.e. Sechach) because this is destroying an Ohel. The same applies to what is near it (i.e. extra wood to make the Sechach thicker; it is Batel to the Sechach)!

Answer #1 (Rav Yehudah): One may take from what is near the walls.

Answer #2 (Rav Menasiya): One may even take from the Sechach, i.e. untied bundles of reeds put on top (they are not Batel to the Sechach).

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 stipulation.

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 stipulation help? "Chag ha'Sukos...la'Shem" forbids Atzei Sukah!

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

Question: This implies that a stipulation does not help for a Sukah for the Mitzvah. This is unlike a Beraisa!

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 stipulation.

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;

Wood of the Sukah gets Kedushah, so it is forbidden all seven days.

Question: Why is Sukah different than Esrog?

Rav: If one designated seven Esrogim, one for each day of Sukos, each day he fulfills the Mitzvah with one and he may eat it immediately;

Rav Asi: Each day he fulfills the Mitzvah with one and may eat it the next day.

Answer: The Mitzvah of the four Minim does not apply at night, so each day is a separate Mitzvah. The Mitzvah of Sukah applies day and night; all seven days are like one long day.

Rishonim

Rif (Beitzah 17a, according to the Ran): The Gemara answers that a stipulation 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 stipulation 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 did not Mevatel them. The Sukah itself is Kodesh because of the verse. Also, the stipulation 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, 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 because of 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 stipulation (it does not help).

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 because of Muktzeh.

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): 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 (R. Tam, brought in Rosh, ibid.): Atzei Sukah needed to make the Sukah Kosher are forbidden mid'Oraisa. More than this is forbidden mid'Rabanan.

Tif'eres Shmuel (7): The Tur says that the Rosh holds that a stipulation helps for Noyei Sukah or extra Sechach even while the Sukah stands. This is like R. Tam.

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.): 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.

Ran (DH u'Mai): The Isur of Esrog is only mid'Rabanan, because it was designated for the Mitzvah. We asked from it because Rabbinic enactments are made similar to Torah law.

Rosh (Beitzah 4:3): A stipulation not to refrain at any time during Bein ha'Shemashos helps for Noyei Sukah. It does not help for Atzei Sukah, for they become Kodesh and they are forbidden all seven days.

Poskim

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

Beis Yosef (DH Kosav b'Orchos): A stipulation helps for Tzuras ha'Pesach or other additions to the walls to beautify them.

Mishnah Berurah (4): One may benefit from them as long as he does not Mevatel Kedushas Sukah from them. E.g., he may lean on them or rest something on them.

Rema: One may not even take a chip for a toothpick.

Shulchan Aruch (ibid.): The wood is forbidden on the eighth day because it is Muktzeh the entire seventh day, including Bein ha'Shemashos.

Mishnah Berurah (2): Since in Chutz la'Aretz we eat in the Sukah on the eighth day, the wood is Muktzeh all of Bein ha'Shemashos after the eighth day, hence it is forbidden on the ninth. If Shabbos is after Simchas Torah it is forbidden on Shabbos, because Yom Tov cannot prepare for Shabbos.

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

Bi'ur Halachah (DH Etzim): If the Sechach was so thick than rain would not fall through, even those who disqualify the Sukah agree that the Sechach is forbidden. One may tell a Nochri to remove the excess Sechach, for mid'Oraisa this is not Stiras Ohel.

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 stipulation to use the excess over Shi'ur Sukah helps for after it falls.

Shulchan Aruch (ibid.): If one made what is needed for a Sukah and later added a wall, that wall is permitted. If one made four walls without specifying (which is the extra one), all are forbidden.

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

Magen Avraham (3): Regarding Tefilin, Hazmanah forbids! That is because Kedushah (of Kisvei ha'Kodesh) is more stringent than standard Mitzvos.

Mishnah Berurah (9): Sitting in it once forbids only if he intended to use it throughout Sukos. If one uses a Sukah year after year, some say that it is forbidden once Sukos comes, even before using it. It is good to be stringent.

Kaf ha'Chayim (22): Sitting in it l'Shem Mitzvah forbids it the entire Sukos.

Kaf ha'Chayim (4): If a Sukah is used year after year, Mahari Molcho forbids it the entire year.