Gemara

Whenever Rav Acha and Ravina argue, Ravina is lenient and Rav Acha is stringent, with only three exceptions - raw meat, Beitzim, and veins;

The Halachah always follows the lenient opinion.

Shabbos 44a (Beraisa #2 - R. Yehudah): One may move a new Ner, but not a used Ner;

R. Meir permits moving any Ner, except for one that was burning on Shabbos;

R. Shimon permits any Ner, except for one burning on Shabbos.

If it extinguished, one may move it, but one may not move a cup, bowl or lantern (even after it extinguished).

(Mar Zutra): Its last clause is like R. Shimon. He permits a lamp, for it holds little oil, so one anticipates that (it will extinguish, and) he will move it. A cup, bowl or lantern holds much oil.

47a (R. Yochanan): Rebbi permitted R. Rumnus to move an incense pan with ashes in it.

(Abaye): This is just like one may remove a Geraf Shel Re'i (a Kli Cheres for excrement).

Objections (Rava): A Geraf Shel Re'i disgusts people, but ashes do not! A Geraf Shel Re'i is exposed, but the ashes are covered!

122b (Mishnah): One may take a hammer to break nuts, an axe to break a ring of figs (it is hard and thick)...

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

R. Nechemyah permits only for a need.

(Rabah): R. Nechemyah permits even Kli she'Melachto l'Heter only to use it, but not for its place.

Question (Rav Safra, Rav Acha and Rav Huna): If so, according to R. Nechemyah, how may we move bowls (after finishing the food inside)?

Answer (Rav Safra): One may move it just like a Geraf Shel Re'i.

157a (Rav Acha or Ravina): For all laws of Shabbos, the Halachah follows R. Shimon, except for Muktzeh Machmas Mi'us (something that disgusts a person), e.g. a used Ner (lamp);

(The other of Rav Acha and Ravina): The Halachah follows R. Shimon even regarding Muktzeh Machmas Mi'us, but not regarding Muktzeh due to Isur, e.g. a Ner in which a fire was burning at the start of Shabbos.

Rishonim

Rif and Rosh (Shabbos 21a and 3:20): The Halachah always follows the more lenient of Rav Acha and Ravina. Therefore we permit Muktzeh Machmas Mi'us, but not Muktzeh due to Isur.

Rambam (Hilchos Shabbos 25:10): One may not move any Kli that was Muktzeh due to Isur, e.g. a Ner that was burning on Shabbos and the candelabra on which the Ner was, or a table that had coins on it. Even if the Ner extinguished or the coins fell off, one may not move it, for any Kli that was forbidden to move Bein ha'Shemashos, one may not move it the entire Shabbos, even if the reason for the Isur went away.

Rambam (11): If a Kli was Muktzeh Machmas Mi'us, e.g. an old kerosene lamp or Kelim for excrement, one may move them on Shabbos if he needs them.

Rosh (Berachos 2:15): Chachamim forbade an Avel in the seven days of mourning to wash for pleasure. They did not decree about an Istnis (finicky person) who is pained if he does not wash. This is unlike R. Chananel, who says that an Istnis is in danger if he does not wash; not everyone who says that he is an Istnis is believed, until experts say so.

Ma'adanei Yom Tov (1 and Divrei Chamudos 61): R. Chananel says that not everyone who says that he is an Istnis is believed, until experts say so. The Tur (YD 381) says that Rav Hai Gaon does not require experts to say so. The Ramban says that it does not depend on danger, but indeed, not everyone who says that he is an Istnis is believed. We must know that he is finicky.

Ran (Shabbos 21a DH v'Ha): We forbid moving a Ner that was burning during Shabbos only if one did not stipulate. If one stipulated to move it after it extinguishes, he may do so, even if it is a big Ner. The Ramban proves so from the Yerushalmi. It says that R. Shimon permits even without a Tanai, but says that regarding a cup, bowl or glass lantern, he forbids even after they extinguish, but if he stipulated, it is permitted. Since we permit Muktzeh Machmas Mi'us, a Tanai helps whether it is of metal or earthenware.

Mordechai (Beitzah 642): One may not move a Ner that was burning during Shabbos even l'Tzorech Gufo u'Mkomo (if one needs to use the Ner or its place for Heter). Some permit, for this is like Geraf Shel Re'i. This is unreasonable. Every fool will say 'I am an Istnis.' It is permitted only for one who is truly an Istnis. One who is stringent does not lose.

Tosfos (Shabbos 44a DH sheb'Ner): Why does R. Yehudah forbid moving a Ner that extinguished? Why is it different than a flimsy Sukah? A Tanai helps for a flimsy Sukah (to use it after it falls), for one expects it to fall. Likewise, one expects a Ner to extinguish! Also according to R. Shimon, a Tanai should permit using the oil that drips from a burning Ner! I answer that the Sukah already existed. Now, Bein ha'Shemashos (twilight of Erev Shabbos; we are unsure at which moment in Bein ha'Shemashos Shabbos begins), he does not refrain from it, and waits for it to fall. A Ner is Hukzteh (put in a situation in which he cannot use it) primarily Bein ha'Shemashos. He actively made it unusable. This is more stringent, therefore a Tanai does not help.

Tosfos (Shabbos 36a DH Ha R. Shimon): R. Shimon forbids only Muktzeh due to Chisaron Kis (something valuable that one uses only offer its special use). He permits (other Muktzeh) only l'Tzorech Gufo u'Mkomo. He admits that one may not move it from the sun to the shade (just to prevent it from being ruined). A Mishnah permits taking a hammer or axe only to break nuts or a ring of figs, but not to move it to the shade. We establish the Mishnah like R. Shimon (Sanhedrin 84b), since the Seifa permits a needle in order to remove a splinter. Also, the Gemara (124b) said that it would be astounding if R. Elazar permitted from the sun to the shade. If R. Shimon permits, it is not astounding if R. Elazar rules like him!

Poskim

Shulchan Aruch (OC 279:2): One may not move a Ner (that was burning during Shabbos) even l'Tzorech Gufo u'Mkomo. Some permit, but this is unreasonable.

Beis Yosef (DH v'Chasav ha'Ran): Also the Rashba permits, and also the Magid Mishneh citing the Ramban and Rashba. R. Yerucham says that the primary opinion holds that a Tanai does not help. Tosfos agrees. The Rivash (93) questioned the Ramban. Since it was forbidden Bein ha'Shemashos, a Tanai should not help! However, he concluded 'the elder ruled' (so we cannot forbid). The Rashbash (407) said that his father (the Rashbatz) answered the question. Something Muktzeh Bein ha'Shemashos is Muktzeh for all of Shabbos only when it itself is Muktzeh, e.g. a Sukah or its wood. A Ner is Muktzeh only due to the flame burning in it. This does not contradict the Rambam, who forbids after it extinguishes, for he did not discuss one who stipulated. Those after the Ramban accepted his words; this is our custom.

Beis Yosef (ibid.): Orchos Chayim says that one must stipulate close to the time of Bein ha'Shemashos. L'Halachah, since Muktzeh is mid'Rabanan, we can rely on those who permit through a Tanai.

Taz (2): Sefer ha'Terumah (308) would not forbid, for he forbids due to Bosis (something that held Muktzeh at the start of Shabbos) only if he intended it to be there for all of Shabbos. We insist that the Neros be on the table only at night. Therefore, it is like one who forgot (and left Muktzeh on Heter). It is not forbidden the entire day (277:3); after it extinguishes, one may move it l'Tzorech Gufo u'Mkomo. The Poskim that the Beis Yosef brought discuss one who intends for the Ner to remain there the entire day. I say that there is no argument. The Mordechai (Shabbos 315) and Agu4 (454) discuss one who intends to remove it after the night.

Bi'ur Halachah (DH Afilu): Even Sefer ha'Terumah would forbid here, for he does not intend to remove the Ner, even through a Nochri, until after it finishes burning. Therefore, it became a Bosis. This is even for a wax candle. All agree that an oil lamp is forbidden, since one could not move it, lest he extinguish (277:3).

Kaf ha'Chayim (7): Many argue with this Heter of the Taz. He himself (309:1) says that we cannot be lenient against the Shulchan Aruch and Poskim. If damage could result, all permit to move it.

Gra (DH Ner): The latter opinion considers it Kli she'Melachto l'Isur. The first opinion holds that they argue only about l'Tzorech Gufo u'Mkomo, for even R. Shimon forbids from the sun to the shade. Tosfos and Hagahos Ashri say so.

Mishnah Berurah (4): The Ner (cup) that holds the oil is forbidden, for it was a Bosis for the flame. The candelabra that holds the Neros is forbidden, for it was a Bosis for the Neros.

Rema: Some say that if one is an Istnis and it is repulsive to him, he may remove it, since for him it is like a Geraf Shel Re'i.

Beis Yosef (DH v'Chasav ha'Mordechai): The Mordechai and Agur permit l'Tzorech Gufo u'Mkomo. The Rambam and Hagahos Ashri forbid. It seems that R. Yerucham and the Rosh (Teshuvah 22:8) agree. A Kli she'Melachto l'Isur (its primary use is forbidden on Shabbos) was not Muktzeh Bein ha'Shemashos, for one could move it l'Tzorech Gufo u'Mkomo. If a Ner was burning Bein ha'Shemashos, one could not move it at all, even for the sake of its use or place. If a rock was on a barrel Bein ha'Shemashos, it was a Bosis for Muktzeh, so it is forbidden the entire day.

Rema: One who is stringent does not lose.

Taz (3): The Bach says that this should say 'one who is stringent will be stringent. One who is lenient does not lose.' If so, why did the Mechaber write in Sa'if 6 'Muktzeh Machmas Mi'us is permitted'? In any case it is permitted because it is like Geraf Shel Re'i! Rather, even though it is repulsive, it is not like Geraf Shel Re'i.

Mishnah Berurah (5): The Bach and Magen Avraham permit even one who is not an Istnis, if it bothers him. It seems that the Bach is lenient only for an earthenware or wooden Ner, which is surely repulsive, but not for metal. Derech ha'Chayim permits only an Istnis, like the Rema.

Kaf ha'Chayim (8): Also the Beis Yosef (309) forbids for one who is not an Istnis. We follow this. If an Istnis wants to be stringent, even though he is pained, he is not called a Hedyot (commoner, i.e. this is praiseworthy).

Mishnah Berurah (6): The Rema permits if he cannot bear that it is there after it extinguished, or during the day.

Mishnah Berurah (7): It is permitted only if it is where he sits. If not, all forbid even for an Istnis.