Gemara

32a - Beraisa #1: One may burn Kelim (for fuel or a wick), but not broken Kelim.

Beraisa #2: Just like one may burn Kelim, he may burn broken Kelim.

Beraisa #3: One may not burn Kelim or broken Kelim.

Resolution: Beraisa #1 is R. Yehudah (who forbids Muktzah and Nolad), Beraisa #2 is R. Shimon (who permits), and Beraisa #3 is R. Nechemyah (who permits Kelim only for their intended purpose).

33a - Rava: If a poker (used to stoke coals) broke one may not burn it on Yom Tov. One may burn Kelim, but not broken Kelim.

Inference: Rava holds like R. Yehudah.

Shabbos 29a - Rav: R. Yehudah permits burning a Keli, with dates (since they are fit to eat, one may move them for other purposes), and with nuts;

He forbids using a Keli that broke (on Yom Tov), or pits or shells left from dates or nuts (that were eaten on Yom Tov, for all of these are Nolad).

R. Shimon permits all of these.

We must teach all three cases:

Had we taught only about the broken Keli, one might have thought R. Yehudah considers it Nolad for it used to be whole, but he would permit pits, for they did not change;

Had we only taught only about the pits, one might have thought that R. Yehudah considers them Nolad because they were hidden and now they are exposed, but he would permit shells, which were exposed from the beginning.

Question (Rav Shmuel bar bar Chanah): R. Yehudah forbids burning a broken Keli. How can he permit a whole Keli? Once one starts to burn it, it is a broken Keli, he moves Isur!

Answer #1 (Rav Yosef): He does like Rav Masnah:

Rav Masnah: If wood fell from a date tree into an oven on Yom Tov, one may add to it a larger amount of wood that was Muchan (to Mevatel the Nolad) and burn all of it.

Answer #2 (Rav Hamnuna): The case is, the wick was a rag less than three by three (Tefachim, so it was never a Keli).

124b - Mishnah: If a Keli 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 says, the fragments may be moved only if they have a use like the original function of the Keli, e.g. if one can pour thick porridge into pieces of a bowl, or pour oil into pieces of glass.

Gemara - Rav Yehudah: They argue about when it broke before Shabbos. R. Yehudah requires a use like the original Keli, 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) as part of the Kelim.

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

Question: When did the Kelim break? If they broke before Yom Tov they are like pieces of wood, surely they may be burned!

Answer: They broke on Yom Tov (and nevertheless, 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 Keli);

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

Rishonim

Rif and Rosh (4:12): If a poker broke one may not burn it on Yom Tov, for we may burn Kelim, but not broken Kelim.

Ran (DH Dorash): In Shabbos (29a) we ask that once one starts to burn a Keli, it becomes a broken Keli. The Gemara concludes that he must add a majority of wood, so the majority will be Heter.

Rif and Rosh (Shabbos 48a and 17:5): In the Mishnah (124a), they argue about when it broke on Shabbos. Chachamim say that the fragments were Muchanim. R. Yehudah says that they are Nolad. If it broke before Shabbos all permit, 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 hold that this rule is only when R. Meir explicitly argues. When there is a Stam Mishnah (even though we assume that it is like R. Meir), the Halachah follows the Stam. This applies even though they explicitly argue in a Beraisa.

Ran (DH Gemara): The Halachah follows Chachamim, who say that it suffices for the fragments to have some use.

Rambam (Hilchos Yom Tov 2:12): One may not burn broken Kelim on Yom Tov because they are Nolad. One may burn whole Kelim or Kelim that broke from Erev Yom Tov because they are Muchanim.

Magid Mishnah: The Gemara says that once he starts to burn it it is a broken Keli and it is forbidden to move it. It answers that (to move it) he must Mevatel it in a majority of Heter. This is why the Rashba forbids moving the Keli after it catches fire. The Rif and Rambam do not mention this; perhaps they had a different text in the Gemara.

Hagahos Ashri (Shabbos 29a): R. Yehudah permits burning a whole Keli on Yom Tov, but not a broken Keli. R. Shimon permits even a broken Keli. If wood fell from a date tree into an oven on Yom Tov, one may add to it a larger amount of Muchan wood and burn all of it.

Poskim

Shulchan Aruch (OC 501:6): If Kelim broke on Yom Tov one may not burn them because they are Nolad.

Mishnah Berurah (22): If the fragments can be used similar to the original Keli, it is as if it is intact.

Magen Avraham (12): One may not move them or even burn them in their place, for this is Hana'ah from Muktzah. Even though when burning intact Kelim they break, the Hana'ah comes afterwards automatically.

Mishnah Berurah (23): Burning a Muktzah Keli is considered like moving it.

Rema: If one burns (intact) Kelim he may not move them after they burned partially, for then they are broken Kelim, unless he added a larger amount of Muchan wood.

Shulchan Aruch (ibid.): One may burn intact Kelim or Kelim that broke before Yom Tov.

Re'em (brought in Magen Avraham 13): This implies that even total Stirah (destruction) of Kelim is permitted. Since we forbid total Stirah of Kelim, we must say that the Keli is fragile.

Rebuttal (Magen Avraham): One may drop a fragile Keli, but one may not totally break it. Also, one may not burn a Keli that a Nochri carved out on Yom Tov, due to Nolad (Shabbos 29a). Since it is not fragile, it should be forbidden due to Stirah! Rather, since even the Melachah of burning is permitted for Ochel Nefesh, also Stirah is permitted. Even though one may not split a Keli for Ochel Nefesh, here it is all one Melachah. Perhaps one may split a Keli (to burn it), just like one may chop wood. One is liable for Stirah only if it is in order to build, therefore it is totally permitted for any (other) need. One may burn a Keli only if he has no wood, for otherwise he transgresses 'Bal Tashchis' for ruining a Keli.

Mishnah Berurah (28): Even though Kelim are not intended for burning, we hold that Kelim may be used even not for their intended purpose.