Gemara

Sukah 36b: R. Chanina would eat a piece of his Esrog and fulfill the Mitzvah with the remains.

46b - R. Yochanan: The Esrog is forbidden on the seventh day and permitted on the eighth. The Sukah is forbidden even on the eighth day.

Reish Lakish: The Esrog is permitted even on the seventh day.

R. Yochanan holds that the Esrog is Muktzeh for the day; Reish Lakish holds that it is Muktzeh (only) for the Mitzvah.

Question (Rav Papa): According to R. Yochanan, why is Sukah different than Esrog?

Answer (Abaye): Sukah is fit for the Mitzvah during Bein ha'Shemashos (between the seventh and eighth day). If he will eat Bein ha'Shemashos, he must eat in the Sukah, therefore it is Muktzeh Bein ha'Shemashos and the entire eighth day. Esrog is not fit for the Mitzvah Bein ha'Shemashos, so it is not Muktzeh Bein ha'Shemashos and not the entire eighth day.

Levi: An Esrog is forbidden even on the eighth day.

Shmuel's father: It is forbidden on the seventh day and permitted on the eighth day.

Shmuel's father retracted and adopted Levi's opinion. R. Zeira adopted the original opinion of Shmuel's father:

R. Zeira: An Esrog is forbidden all seven days (even) if it became Pasul.

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 he may eat it the next day.

Rav Asi holds like R. Yochanan, that it is Muktzeh for the day. Rav holds like Reish Lakish, that it is Muktzeh for the Mitzvah.

Question: In Bavel, we keep two days of Yom Tov (due to a Safek about which day it is). What is our law?

Answer #1 (Abaye): On the eighth day, which is Safek (perhaps it is really) the seventh day, it is forbidden. On the ninth day/Safek eighth, it is permitted.

Answer #2 (Mereimar): Even on the eighth/Safek seventh day it is permitted.

In Sura, people followed Mereimar. Rav Shisha brei d'Rav Idi followed Abaye. The Halachah follows Abaye.

Rishonim

Rif (22b): The Esrog is forbidden on the seventh day and permitted on the eighth. For us, who keep two days of Yom Tov, it is forbidden on the eighth day/Safek seventh. It is permitted on the ninth day/Safek eighth.

Rif and Rosh (4:5): If one designated seven Esrogim, each day he fulfills the Mitzvah with one and he may eat it the next day. The Halachah follows Rav Asi, because R. Yochanan holds like him.

Rambam (Hilchos Lulav 7:27): It is forbidden to eat the Esrog the entire seventh day. Since it was Muktzeh for the Mitzvah that day, it is Muktzeh the entire day. One may eat it on the eighth day. Nowadays that we keep two days of Yom Tov, even though we do not take it on the eighth day, it is forbidden that day just like when they kept two days due to Safek.

Magid Mishnah: We are not stringent to take the Lulav on the eighth day because it is only mid'Rabanan after the first day. We sit in the Sukah on the eighth day because it is mid'Oraisa all seven days.

Kesef Mishneh: Also, we do not take the Lulav on the eighth day because it is Muktzeh. This does not apply to Sukah.

Rambam (ibid.): If one designated seven Esrogim, each day he fulfills the Mitzvah with one and he may eat it the next day.

Rosh (4:4): R. Yochanan forbids the Esrog on the seventh day and permits on the eighth. The Sukah is forbidden even on the eighth. This is because Sukah is fit for the Mitzvah during Bein ha'Shemashos; if he will eat then, he must eat in the Sukah.

Question: Also the Esrog is Muktzeh Bein ha'Shemashos, for if he was not Yotzei yet that day, he must take it!

Answer #1 (Rashi DH Esrog Lo): We discuss one who was already Yotzei that day.

Answer #2 (Ran DH Omar): Even if he was not Yotzei that day, nowadays Esrog is only mid'Rabanan after the first day, therefore in a Safek (Bein ha'Shemashos) he is exempt. We do not say that since it was forbidden part of the day it is forbidden the entire day (including Bein ha'Shemashos). We say so only regarding Shabbos (and Yom Tov), because something that was not prepared for Shabbos cannot prepare itself on Shabbos. Muktzeh on a Yom Chol is only as long as the reason for Muktzeh applies.

Question: Also the Esrog is Muktzeh Bein ha'Shemashos, for perhaps it is still day!

Answer (Rosh 4,5): The Esrog is Muktzeh Bein ha'Shemashos only because it was Muktzeh the previous day. In such a case it is not Muktzeh on the coming day. Similarly, if one designated seven Esrogim, one for each day, each day he fulfills the Mitzvah with one and he may eat it the next day. It is permitted even though it was forbidden Bein ha'Shemashos of the previous day. Similarly, when Shabbos and Yom Tov are adjacent, an egg born on one is forbidden on the other because of Nolad, but not because it was Muktzeh Bein ha'Shemashos. An egg born on Yom Tov Rishon is permitted on Yom Tov Sheni of Chutz la'Aretz. If one had Ma'aser Tevel in a flask that was a Tevul Yom and stipulated that it become Terumas Ma'aser after dark (when the flask will be totally Tahor), it is invalid for an Eruv only because an Eruv takes effect before dark, or because the food must be permitted before dark. Because the food was Muktzeh Bein ha'Shemashos due to the previous day, it is not Muktzeh on Shabbos. Bein ha'Shemashos beginning Shemini Atzeres, one who eats must eat in the Sukah only because perhaps it is still the seventh day. Nevertheless, the Sukah is Muktzeh on Shemini Atzeres because one (who wants to eat) must do an action Bein ha'Shemashos.

Rashi says that Levi forbids on the eighth day because it was Muktzeh the previous day. I have proven that this is wrong. Rather, he decrees to forbid Esrog lest people use the Sukah on the eighth day. In Bavel we forbid the eighth day, because if we would find out that Elul was full we would realize that today is really the seventh day.

Ran (DH ul'Didan): It is forbidden lest people think that it is definitely the eighth day, and they will not sit in the Sukah.

Rosh (ibid.): The custom is not to use Noyei Sukah on Shabbos which is right after Simchas Torah. Really, it is permitted, like I explained. The Halachah is that Shabbos and Yom Tov are two Kedushos. R. Peretz forbids an Esrog on Simchas Torah which is on Motza'ei Shabbos due to Hachanah (Shabbos may not prepare for Yom Tov). The Halachah is that the Esrog is permitted on the ninth day, even if it is Motza'ei Shabbos.

Hagahos Maimoniyos (7:2): Hachanah does not apply to the Esrog, for nothing was done to it. No Tana or Amora forbids an Esrog on the ninth day.

Poskim

Shulchan Aruch (665:1): The Esrog is forbidden (even) on the seventh day, for it was Huktzeh (set aside) for all seven days.

Shulchan Aruch (ibid.): This is even if it became Pasul after using it for the Mitzvah. It is permitted on Shemini Atzeres, but not in Chutz la'Aretz.

Beis Yosef (DH Kasuv b'Orchos Chayim): Orchos Chayim learns from the Yerushalmi that one may eat an Esrog that became Pasul on the first day: This is difficult. Bein ha'Shemashos it became Muktzeh for the entire day!

Taz (2): The Yerushalmi is like Reish Lakish, who says that it is Muktzeh only for the Mitzvah. The Beis Yosef questioned the Orchos Chayim, for the Halachah follows R. Yochanan. Perhaps the Orchos Chayim cites the Yerushalmi to teach that if it became Pasul, it is permitted on the eighth day (in Chutz la'Aretz), even though normally it is forbidden lest people be lenient about Sukah.

Kaf ha'Chayim (7): The Shulchan Aruch and others forbid on the eighth day even if it became Pasul. One should not be lenient.

Shulchan Aruch (ibid.): In Chutz la'Aretz it is forbidden on Shemini Atzeres and permitted on Simchas Torah, even if Simchas Torah is after Shabbos. Some forbid when it is after Shabbos.

Magen Avraham (2): Hachanah does not apply to it, for it was already in the world and nothing was done to it.

Shulchan Aruch (2): If one designated seven Esrogim, each day he fulfills the Mitzvah with one and he may eat it the next day. It is forbidden the same day because it was set aside for the entire day.

Beis Yosef (DH Hifrish): The Ro'oh says that since he designated one for each day, it is like explicitly saying it is set aside only for the Mitzvah, therefore it is not Muktzeh Bein ha'Shemashos.

Magen Avraham (3): We must say so only according to the opinion that permits eating it the same day.

Mahari Avuhav (brought in Beis Yosef OC 21 DH v'Al): Letter of the law one may eat from an Esrog during Sukos as long as he leaves the Shi'ur of k'Beitzah so it will still be Kosher for the Mitzvah. However, it is not proper to do so.

Rebuttal (Beis Yosef, ibid.): More than the Shi'ur needed for the Mitzvah is not Muktzeh for the Mitzvah. L'Chatchilah one may eat the excess. If it was not proper to do so, R. Chanina would not have done so! The She'altos forbids eating from an Esrog if a Shi'ur will not remain.

Defense (Drishah DH Kosav and Prishah 4): Mahari Avuhav discusses when the Esrog was designated for all of Sukos. If it was Muktzeh only for one day, all agree that l'Chatchilah one may eat from it as long as he leaves a Shi'ur.

Note: Seemingly, one should use the most Mehudar Esrog he can find all seven days! Also, if something was used for a Mitzvah, we prefer to use it for another Mitzvah (e.g. Eruv Tavshilin for Lechem Mishneh, the same Lechem Mishneh for all three Shabbos meals)! Perhaps R. Chanina thought that it was more important to demonstrate the Halachah. Alternatively, every day it became discolored through being held (especially if he allowed many others to use it), so it was more Mehudar to use a fresh Esrog each day.