Gemara

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).

Abaye: They argue about an Eli, but all permit a Tavar Garmi (a board special for chopping meat).

Objection: This is obvious! The Mishnah teaches the argument (only) about an Eli!

Version #1 - Answer: One might have thought that they argue even about Tavar Garmi, and the Mishnah taught Eli to teach that Beis Hillel permit even though it is a Keli she'Melachto l'Isur. Abaye teaches that this is not so.

Version #2 - Answer (Abaye): My Chidush is that all permit even a new Tavar Garmi. One might have thought that we are concerned lest he take it and not chop. I teach that this is not so.

Shabbos 43a - R. Yitzchak: Just like one may not put a Keli under a hen to catch an egg (that will not laid on Shabbos), one may not put a Keli over an egg (laid on Shabbos) to guard it.

He holds that a Keli may be moved on Shabbos only for the sake of something that may be moved on Shabbos.

Question - Beraisa: An egg laid on Shabbos or Yom Tov may not be used to cover a jar or to support the leg of a bed, but one may put a Keli over it to guard it.

Answer: The case is, he needed the place of the Keli.

123b - Beraisa: At first, they said that only three Kelim may be moved - a knife to cut a ring of figs, a big spoon to remove froth from pots and a small table knife. (This does not include Kelim that hold things. Surely, cups, bowls and plates were permitted! - Tosfos)

They permitted more, and permitted more, until they permitted all Kelim except for a big saw or a plow blade.

Question: What does it mean 'They permitted more, and permitted more'?

Answer #1 (Abaye): They permitted to use a Keli she'Melachto l'Heter, then permitted to move it when its place is needed, then permitted to use a Keli she'Melachto l'Isur (for permitted uses) but not for the sake of its place;

Objection (Rava): 'They permitted' connotes for its use and for its place!

Answer #2 (Rava): They permitted a Keli she'Melachto l'Heter both for its use and place, then permitted to move it out of the sun to the shade. Later, they permitted a Keli she'Melachto l'Isur for its use or place, but not from the sun to the shade;

Still, only Kelim that can be moved by one person were permitted, not those that require two, until they permitted all Kelim, even those that require two people.

124a - Question (Mishnah): (On Yom Tov) one may not support a bowl or door with a piece of wood.

A piece of wood is Melachto l'Heter on Yom Tov (it may be burned). This teaches that a Keli she'Melachto l'Heter may not be moved for its use or place!

Answer: There, since it is Melachto l'Isur on Shabbos, we decree to forbid on Yom Tov like on Shabbos.

Question: Even on Shabbos it should be permitted. Melachto l'Isur is permitted for the sake of its use or place!

Answer: That is only if it is a Keli. This is not a Keli.

Rishonim

Rif (Shabbos 48a) and Rosh (Shabbos 17:4): All Kelim may be moved for a need (or without a need). Rava explains that this permits a Keli she'Melachto l'Heter even from the sun to the shade, and a Keli she'Melachto l'Isur for its use or place, but not from the sun to the shade.

Ran (Beitzah 2a DH u'Minah): If one started to move a Keli she'Melachto l'Isur for its place, he need not put it down right next to where he needs. Rather, since he was allowed to pick it up, he may put it down anywhere he wants. We learn from R. Yitzchak (Shabbos 43a). It is unreasonable to say that he establishes the Beraisa to discuss one who needed a Keli that was right next to the egg.

Note: The Ran permits to put it down where he wants because he was allowed to pick it up. This should also apply if he picked it up to use its place. Perhaps the Ran addresses only one case because it is obvious that the same applies to the other. Tosfos (Beitzah 3b DH Aval) explicitly permits one who moved a Keli she'Melachto l'Isur for its use or place to put it anywhere he wants.

Rambam (ibid.): Any Keli she'Melachto l'Isur, whether it is made of wood, Cheres, stone or metal, may be moved for its use or place, but not for the sake of the Keli.

Poskim

Shulchan Aruch (OC 308:3): One may use a Keli she'Melachto l'Isur for its use, e.g. a smith's hammer to break open nuts or an axe to cut a ring of dried figs, or to use its place.

Mishnah Berurah (10): If the majority usage is for Isur, it is considered Melachto l'Isur.

Kaf ha'Chayim (19): Even a new Keli is forbidden if it will be used for Isur, even an Isur mid'Rabanan.

Mishnah Berurah (12): One may not use a Keli she'Melachto l'Isur if he could use a Keli she'Melachto l'Heter.

Kaf ha'Chayim (21): One may move Muktzah Machmas Mi'us (something offensive) to use it or its place.

Shulchan Aruch (ibid.): One may put it down wherever he desires.

Magen Avraham (7): Even if one picked it up by mistake (without a Heter), he may put it anywhere he wants.

Mishnah Berurah (13): The Gra (266:12, brought below) disagrees. The Heter to place Muktzah where one wants applies only to Keli she'Melachto l'Isur. However, moving through the body is not considered Tiltul. Therefore, one may kick Muktzah with his feet.

Chazon Ish (47:13): One may not kick Muktzah with his feet. This is proper Tiltul!

Shulchan Aruch (ibid.): It is forbidden to move it from the sun to the shade, i.e. he does not need it, he just wants to prevent it from being broken or stolen.

Mishnah Berurah (15): This is permitted through a Nochri.

Magen Avraham (8): The Yerushalmi permits to intend to use it for Heter in order to move it from the sun to the shade.

Rema (266:12): If on Shabbos one mistakenly was wearing his wallet in his house, he may go to a room and loosen his belt to make the wallet fall in a concealed place.

Rebuttal (Gra DH u'Mi): The Rema learns from Shabbos 43a, but this is wrong. There it was permitted to move the Keli from the beginning, so he may place it where he wants.