Gemara

11a - Mishnah - Beis Shamai: One may not take an Eli (a thick board used for crushing wheat) to chop meat on it (on Yom Tov);

Beis Hillel permit.

Beraisa: Both agree that if one chopped meat on it, it may not be moved (afterwards).

10a - Question: The Mishnah of Eli shows that Beis Hillel are more lenient about Simchas Yom Tov than Beis Hillel. Regarding Shechitah of Chayos and birds (2a), Beis Shamai are more lenient!

Answer #1 (R. Yochanan): The opinions must be switched!

Answer #2: The opinions need not be switched. Beis Shamai permit Shechitah only when a shovel is inserted in dirt. Beis Hillel permit the Eli because it is considered a Keli.

Shabbos 122b (Mishnah): One may take a hammer (to break nuts).

Rabah: The Mishnah permits a smith's hammer to break nuts.

Rabah permits moving a Keli she'Melachto l'Isur if it itself is needed.

Question (Abaye - Beraisa - Beis Shamai): One may not take an Eli to chop meat on it (on Yom Tov);

Beis Hillel permit.

Both agree that if one chopped meat on it, it may not be moved (afterwards).

Intended answer (Rabah): This is like R. Nechemyah, who permits moving a Keli only for its intended purpose.

Answer (Rabah): Rav Chinena said that all agree about Sikei, Zairei and Mezorei (special tools for dying, making or laundering garments). Since people care about them, they fix a place for them (so they may not be used for other purposes). The same applies to an Eli.

124a - Mishnah: All Kelim may be moved for a need or without a need;

R. Nechemyah says, they may be moved only for a need.

Rava: 'For a need' is a Keli she'Melachto l'Heter for its use or place. 'Without a need' is from the sun to the shade, or a Keli she'Melachto l'Isur for its use or place, but not from the sun to the shade;

R. Nechemyah says that even a Keli she'Melachto l'Heter may be moved only for its use or place, but not from the sun to the shade.

Question (Abaye - Beraisa): Both (Beis Shamai and Beis Hillel) agree that if one chopped meat on it (an Eli), it may not be moved.

Answer (Rava): That refers to moving it from the sun to the shade.

Rishonim

Rif and Rosh (1:15): Beis Shamai and Beis Hillel agree that if one chopped meat on an Eli, it may not be moved. This refers to moving it from the sun to the shade, but for its use or place is permitted, like the Gemara says in Shabbos.

Ran (DH Masnisin): Even though Beis Shamai also permit a Keli she'Melachto l'Isur for its use or place, they forbid an Eli to chop meat because it is a weekday action.

Question (Hagahos Chavos Ya'ir 1): The Rif (Sof Beitzah) forbids all Muktzah on Yom Tov!

Answer (Hagahos Chavos Ya'ir): Here, the Rif means that the Eli may be used on Shabbos.

Note: The Rif says only that the Halachah follows R. Yehudah regarding Yom Tov. It appears that even R. Yehudah permits a Keli she'Melachto l'Isur for its use or place. The only Tana who forbids is R. Nechemyah (Shabbos 124a).

Rosh (ibid.): The Ri forbids an Eli even for its use or place, because it is compared to Sikei, Zairei and Mezorei (Muktzah Machmas Chisaron Kis, i.e. it is valuable, and people fix a place for it - Shabbos 123a). The Yerushalmi supports this. The Rif holds like Rava's answer there, that the Beraisa forbids moving it from the sun to the shade.

Korban Nesan'el (2): Often, the Rif does not rule like Abaye or Rava against his Rebbi. He is unsure about whether or not the Halachah follows one of them against his Rebbi. Here it is a mid'Rabanan matter, so the Rif is lenient.

Beis Yosef (OC 499 Sof ha'Siman): R. Yerucham says that the Ri forbids any Keli she'Melachto l'Isur for its use and place. This is wrong. The Ri forbids only Kelim such as an Eli, for which people designate a place.

Bartenura (Beitzah 1:5): Beis Hillel permit to use the Eli for the sake of Simchas Yom Tov, even though it does not have Toras Keli.

Rebuttal (Tosfos Yom Tov): The Gemara (10a) says that an Eli has Toras Keli! Rather, the Gemara (Shabbos 123a) says that an Eli is Muktzah Machmas Chisaron Kis. Therefore, normally it is forbidden even for its use; Beis Hillel permit because of Simchas Yom Tov.

Poskim

Shulchan Aruch (OC 499:5): An Eli, i.e. a thick heavy board, is a Keli she'Melachto l'Isur; it is used to crush wheat. One may move it to chop meat on it (on Yom Tov). Afterwards one may not move it from the sun to the shade.

Mishnah Berurah (18): (It is Melachto l'Isur because) one may not crush wheat on Yom Tov.

Magen Avraham (11): Even though it is Muktzah Machmas Chisaron Kis, one may use it for the sake of Simchas Yom Tov. Even though the Rif explains the Gemara differently, he agrees to this.

Rebuttal (Bi'ur Halachah DH Ela): The Rif explains that the Eli is a regular Keli she'Melachto l'Isur, so there is no source to permit Muktzah Machmas Chisaron Kis for the sake of Simchas Yom Tov. The Rambam does not discuss this. He relies on the general rules he gave in Hilchos Shabbos, for Yom Tov is no more lenient. Also the Rashba seems to hold like this.

Kaf ha'Chayim (36): It is permitted even though chopping on it makes grooves in it.

Rema: It is permitted for the sake of using it or its place.