Gemara

(Mishnah): If a Kli may be moved, its fragments may be moved with it, on condition that they have a use, e.g. pieces of a bowl to cover a barrel, or pieces of glass to cover a flask;

R. Yehudah permits only if they have a use like the original function of the Kli, e.g. if one can pour thick porridge into pieces of a bowl, or pour oil into pieces of glass.

(Rav Yehudah): They argue about when it broke before Shabbos. R. Yehudah requires a use like the original Kli, and Chachamim accept any use;

If it broke on Shabbos, all permit, for at the start of Shabbos the fragments were Muchanim (prepared for use) Agav Avihem (as part of the Kelim they came from).

Rejection (Rav Zutrai - Beraisa): One may burn Kelim (on Yom Tov), but not broken Kelim.

If the Kelim broke before Yom Tov they are like pieces of wood. Surely one may burn them!

Rather, they broke on Yom Tov. (Even so, one may not burn them!)

Correction: Rather, Rav Yehudah taught that they argue about when it broke on Shabbos. Chachamim say that the fragments were Muchanim. R. Yehudah says that they are Nolad (unless they can be used like the original Kli);

If it broke before Shabbos all permit, for the fragments were Muchanim for use before Shabbos.

Rishonim

Rif and Rosh (48a and 17:5): In the Mishnah, they argue about when it broke on Shabbos. Chachamim say that the fragments were Muchanim. R. Yehudah says that they are Nolad. All permit if it broke before Shabbos, for the fragments were Muchanim for use before Shabbos. BaHaG cites Rav Tzemach Gaon, who applies here the rule that the Halachah follows R. Yehudah against R. Meir. We apply this rule only when R. Meir explicitly argues. The Halachah follows a Stam Mishnah [even though we assume that it is like R. Meir], even if they explicitly argue in a Beraisa.

Rambam (Hilchos Shabbos 25:12): If a Kli may be moved, if it broke on Shabbos or before Shabbos its fragments may be moved with it, on condition that they have a use like the original function of the Kli, e.g. pieces of a bowl to cover a barrel, or pieces of glass to cover a flask, and everything similar. If the fragments cannot be used for any Melachah, one may not move them.

Rosh (Teshuvah 22:8, 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.

Beis Yosef (OC 308 DH v'Chasav): The Rosh permits lifting the table even with nut shells that animals cannot eat, for they are Batel to the bread. Semag forbids what is not proper for animals to eat.

Poskim

Shulchan Aruch (OC 308:6): Any Kli that broke, even on Shabbos, one may move its fragments, as long as they have a use.

Beis Yosef (DH Kol): The Rif, Rambam and Rosh rule like the first Tana.

Taz (5): When we establish that the Tana'im argue about Kelim that broke on Shabbos, the Mishnah that forbids lighting with broken Kelim is like R. Yehudah, and the first Tana permits. The Tur (501) explicitly says that R. Yehudah forbids a spit that broke on Yom Tov. This is difficult, for Rava (143a) forbids lighting with broken Kelim. If so, here he should require a use like the initial use, like R. Yehudah, since these laws depend on each other! How can we rule here like the first Tana, against Rava? Also, why does the Rambam rule here like the first Tana, but regarding broken Kelim, he forbids like R. Yehudah? We must say that they hold that these matters do not depend on each other. Only in the Hava Amina, that the Heter is because they are prepared Agav Avihem, this applies also to broken Kelim. We asked that the Mishnah of broken Kelim is not like any Tana. We concluded that they argue about Kelim that broke on Shabbos. The Heter is not because they were prepared Agav Avihem. Rather, the pieces themselves have a use, even though it is not the initial use. R. Yehudah holds that since it is a different use, they are Nolad. If so, all forbid burning broken Kelim, since the shard is not a Kli at all. They argue only when the shard is destined to be used for another task. This is why the Rambam forbids burning broken Kelim.

Question: In Beitzah (33a), Rava forbids burning a broken spit. The Gemara concludes that he holds like R. Yehudah, who forbids Muktzeh!

Answer (Taz): The Rambam holds that the inference is from the Reisha there, that one may not enter a storehouse of wood and take a piece for a spit, for wood is only for burning. If the inference were from the Seifa, the Gemara would have concluded that he holds like R. Yehudah, who forbids Nolad. The Tur forbids burning broken Kelim according to R. Yehudah, for this is clear. It is possible that the first Tana agrees.

Magen Avraham (15): The opinion that forbids Nolad requires a use like the initial use. He also forbids a Kli that a Nochri made on Shabbos.

Bi'ur Halachah (DH l'Shum): Many Acharonim argue with the Magen Avraham, and say that since they are useful for some Melachah, they are not Nolad. They are Muchan. However, if a Nochri made a Kli, it is surely Nolad.

Kaf ha'Chayim (60): The Machatzis ha'Shekel and Rabbeinu Zalman defended the Magen Avraham.

Kaf ha'Chayim (59): We forbid Nolad only on Yom Tov.

Mishnah Berurah (28): Broken Kelim are like pebbles. They are Muktzeh Machmas Gufo (they are not prepared for use). However, since they came from a Kli, the name of the Kli is not Batel from them as long as they are use for some Melachah, e.g. to cover. Even though pebbles can be used to cover, they are not a Kli, like it says in the next Sa'if.

Kaf ha'Chayim (56): If they can cover only a small Kli, but not a barrel, they are not considered Kelim at all.

Kaf ha'Chayim (58): All the more so it is permitted if it broke before Shabbos, for then it was prepared for another Melachah before Shabbos. Even R. Yehudah agrees that this is not Nolad.

Bi'ur Halachah (DH Kol): Mishbetzos Zahav says that this law refers to Kli she'Melachto l'Isur, which one may move for its use or place. Also for them, it suffices if they are useful for some Melachah.

Kaf ha'Chayim (57): The Mishnah discusses "Kelim that may be moved" to exclude Kelim Muktzeh due to Chisaron Kis. (They are valuable. One uses them only for their intended use.) One may not move their shards even for their use or place. If initially a Kli was Melachto l'Isur, and now it is proper for Meleches Heter, or it was Melachto l'Heter, and now it is proper for Meleches Isur, now it is permitted only for its use and place, but not from the sun to the shade.

Shulchan Aruch (ibid.): E.g. pieces of a bowl to cover a barrel, or pieces of glass to cover a flask.

Bi'ur Halachah (DH Kegon): One may move these even from the sun to the shade. If they are not useful for some Melachah, one may not move them even for their use or place.

Shulchan Aruch (ibid.): However, [one may not move them] if they cannot be used for any Melachah.

Kaf ha'Chayim (62, citing Prishah 6 and Eliyahu Rabah 16): One may not move them to save them from being stolen, but one may move them for their use or place, like the Rosh says.

Rema: If they broke in a place where they can damage, e.g. glass broke on the table or in a place where people walk, one may move the pieces to clear them to where they will not damage.

Magen Avraham (16): One may not clear away earthenware [for it does not damage much]. Even though he tramples it with his feet and breaks it, it is Davar she'Eino Miskaven. Also, there is no Isur to break them (Sa'if 44, and 318:12). However, 337:2 connotes that it is forbidden. Perhaps there he overtly breaks, and here it happens by itself.

Mishnah Berurah (30): One may not clear it away with his hands, but one may do so with his feet. One may move Muktzeh with his body.

Kaf ha'Chayim (64): If one can move them Min ha'Tzad (through something else), he should not move them directly.