Gemara

(Beraisa): One may move bones, for dogs eat them.

143a (Mishnah - Beis Shamai): We may remove bones and peels from the table;

Beis Hillel say, we shake the entire Tabla (tabletop, and they fall off).

We remove from the table crumbs less than a k'Zayis and pods of peas and of lentils, for animals eat them.

(Rav Nachman): The opinions in our Mishnah must be switched. Beis Shamai forbid Muktzeh like R. Yehudah, Beis Hillel permit like R. Shimon.

(Mishnah): Pods of peas...

Our Mishnah is like R. Shimon, who permits Muktzeh.

One may not move pits of good dates. (No one feeds good dates to animals, no fruit sticks to the pits, and people do not eat the pits.)

Shmuel would move them along with bread. Rabah would move them along with a flask of water.

Rav Huna brei d'Rav Yehoshua piled them up in front of himself [so they would repulse him], making them like Geraf Shel Re'i (a Kli for excrement, which one may move).

Objection (Rav Ashi): One may not make Geraf Shel Re'i l'Chatchilah!

Rav Sheshes, Rav Papa and R. Zecharyah ben Avkulas would throw them [or spit them out], across the table or in back of the bed [on which they ate].

Rishonim

Rif (59b): Really, Beis Hillel hold like R. Shimon. The bones and peels must be proper for dogs and animals. If not, R. Shimon forbids them.

Ran (DH Gemara): This is even if they separated [from the food] on Shabbos.

Rambam (Hilchos Shabbos 26:16): One may move anything proper to feed to animals and birds, including bones proper for dogs and all peels and pits proper for animals. If it is not proper for animals, one eats the food and casts [the peel or pit] in back of him, and one may not move it.

Magid Mishneh: Some Chachamim would spit the pits in back of themselves, for l'Chatchilah one may not make Geraf Shel Re'i.

Rosh (Teshuvah 22:8): Shmuel permits moving date pits along with bread. In Shmuel's area, animals did not eat them. If they ate them, Shmuel would permit direct movement, for he rules like R. Shimon, who permits Muktzeh. However, if they are not proper for animals, why are they permitted along with bread? R. Tam says that they were proper for animals, and Shmuel was stringent for himself [to move them only with bread]. If not, they are like rocks. One may not move rocks along with bread! One may move only a Kli along with bread.

Rosh (ibid., ha'Derech ha'Hei): One may move bones, peels and date pits with bread, for they are not important, and they are Batel to the bread.

Tosfos (Beitzah 2a DH u'Veis): Why may one shake the table? It is a Bosis for Isur! We can say that he does not intend to leave them there the entire day. Alternatively, since there is also Heter there, it is a Bosis for Isur and Heter. Also, he did not put there intentionally. He is not concerned where they fall.

Poskim

Shulchan Aruch (OC 308:27): One may clear off the table bones proper for dogs, peels proper for animals, and crumbs less than a k'Zayis.

Beis Yosef (DH Atzamos): Rashi says that the Mishnah (143a) discusses hard bones not proper for dogs. Tosfos says that the bones are proper for dogs, for we establish our Mishnah like R. Shimon, and R. Shimon requires that they are proper for animals! The Rif, Rambam and Rosh say so.

Kaf ha'Chayim (179): Shiltei ha'Giborim (59b-60a, 2) says that the custom is to move peels of nuts and almonds that are not proper even for animals, for people rely on Rashi. The Bach says that Rashi agrees that the conclusion forbids.

Magen Avraham (49): The Shulchan Aruch is according to the opinion that permits Nolad. See 495:4.

Kaf ha'Chayim (177): On Yom Tov one may move them only the way we permit for bones and peels not proper for animals [below].

Mishnah Berurah (110): They are permitted even if they came off the meat on Shabbos, so before Shabbos they were not prepared for dogs, and all the more so if they separated before Shabbos.

Mishnah Berurah (111): Because the crumbs are so small, Stam they are destined to be only for animals and birds.

Shulchan Aruch (ibid.): If the peels are not proper for animals, one may not move them. Rather, he shakes the entire tabletop, and they fall off.

Taz (18): Likewise, one may clear them off through something else, e.g. through a knife, since he does not touch them. This is Tiltul Min ha'Tzad (310:8).

Kaf ha'Chayim (181): All the more so one may shake them off, for this is easier to permit than Tiltul Min ha'Tzad.

Magen Avraham (50): Also Tosfos (Shabbos 44b DH Yesh) says that even if he puts on it on Shabbos, one may not shake the Isur. I wrote above (306:14) that it is a Bosis only if the Muktzeh was on it Bein ha'Shemashos. Siman 311:1 connotes like this. Even Tosfos agrees that if the Muktzeh was removed, one may move the table, since the Muktzeh was not on it Bein ha'Shemashos. The Gemara (44b) explicitly says so.

Kaf ha'Chayim (182): This is because there is no Muktzeh for half of Shabbos.

Mishnah Berurah (113): Shells of nuts and eggs and similar matters are not proper for animals.

Mishnah Berurah (114): If a little meat remains on the bones one may move along with the meat.

Mishnah Berurah (115): It is permitted because he does not move them with his hands. Likewise, he may clear them from the tablecloth through something else, e.g. a knife, if he needs the place of the bones and peels. Tiltul Min ha'Tzad is permitted for the sake of Heter. If many gathered, and it is repulsive to him to leave them on the table, he may remove them by hand, for it is like Geraf Shel Re'i to him.

Bi'ur Halachah (DH Mena'er): The Magen Avraham (337:4) permits sweeping them from the ground with a feather. The Mishnah says that one removes the table and shakes it. Milchamos Hash-m explains that he takes it with the peels on it to where the oven is (a lowly place), and shakes it there. It seems that the Shulchan Aruch and Semag forbid moving it with the peels on it. Rather, he shakes them off where it is. These opinions are like the two opinions in Chidushei ha'Rashba (143a DH Ha d'Metaltelei).

Kaf ha'Chayim (184): The Taz (311:2) forbids moving Isur through a feather when it is not in front of him, so it is not Geraf Shel Re'i. Alternatively, there, he does not need the place.

Shulchan Aruch (ibid.): If there is bread on the table, one may lift the table and move it with the peels, which are animal food.

Beis Yosef (DH u'Mah she'Chasav v'Pirurin): The Rosh permits lifting the table even with nut shells that animals cannot eat, for they are Batel to the bread. Semag says that if they are not proper for animals to eat, one may not lift the table, but he may shake the table, just like coins on a pillow. One shakes it, and they fall. This connotes one may not lift it and move it as is [with them on it]. If he needs the place of the table, all permit to lift it and move it as is, just like coins on a pillow. Perhaps Semag does not argue. He discusses nuts shells and bones not proper for animals, and the Rosh discusses when there is bread on the table, therefore one may move them along with the bread.

Magen Avraham (51): Whenever there is a Bosis for Isur and Heter, if one can shake [off the Isur], he must do so (309:3). One may move along with a loaf or child only for a Mes. Here is different, for they are Batel to the bread. Whatever is not Batel to the bread, and is not a Kli, one may not move it with bread (Rosh, R. Yerucham). This connotes that one may put next to bones and peels a loaf or child. However, Piskei ha'Rosh (Beitzah 2:15) permits only throwing with his mouth [when he eats] into the basket with bread, but not to put bread on them in order to move them. The Ramban and Ran say that we permit only a Mes through a loaf or child, i.e. when one moves the Muktzeh. Always, one may move the Muktzeh along with the Heter. Why aren't the bread and basket a Bosis for Isur? Since he puts the pits there only in order to throw them, it is not a Bosis. The Beis Yosef's words are difficult in Teshuvas ha'Rosh.

Machatzis ha'Shekel: Tosfos Shabbos says that the text of the Magen Avraham should say 'their words are difficult', i.e. the Ramban and Ran. I defend our text. The Beis Yosef said that Semag and the Rosh need not argue; the Rosh discusses when there is [already] bread on the table. The Magen Avraham rejected this, for the Rosh permits putting bread on!

Kaf ha'Chayim (185): The Taz (310:8) says similarly, that if Muktzeh was placed on Heter on Shabbos, one may put Heter next to it and move it with both of them, for there is no Muktzeh for half of Shabbos. Tosfos (Beitzah 2a) permits this even if the bones and peels were there from before Shabbos, since he does not care where they fall.

Mishnah Berurah (116): Perhaps one may put bread there l'Chatchilah in order to lift the table, if it is hard for him to shake it.

Shulchan Aruch (ibid.): If he needs the place of the table, even if there are on it only matters improper for animals, one may lift it and move them.

Kaf ha'Chayim (186): I.e. it does not help to shake the table, for they will fall in a place that he needs.

Gra (DH v'Im): Tosfos (Beitzah 2a) says that the bones and peels are considered as if he forgot them, since we permit shaking them.

Shmiras Shabbos k'Hilchasah (16:10): If one normally plays with apricot pits during the week, one may move them. (Mishnah Berurah 308:93 says that regular use during the week permits even regarding stones.)

Shmiras Shabbos k'Hilchasah (20, footnote 82): Surely one may move orange peels, for people sweeten them and eat them. It is not clear whether one may move grapefruit peels.