Gemara

53b (Rav Yehudah): We bless on fire only on Motza'ei Shabbos, for that is when it was created.

An elder, or Rabah bar bar Chanah: Correct! Also R. Yochanan said so.

R. Aba (to Ula): You [and Rabah bar bar Chanah] say in the name of R. Yochanan that we bless on fire only on Motza'ei Shabbos. Surely, this is because that is when it was created!

(Ula and Rabah bar bar Chanah): No, R. Yochanan taught [a different law, i.e.] that when Yom Kipur is on Shabbos, in all places we light to honor Shabbos.

54a: Ula and Rabah hold like R. Binyamin bar Yefes:

(R. Binyamin bar Yefes citing R. Yochanan): We bless on fire on Motza'ei Shabbos and on Motza'ei Yom Kipur.

People follow this opinion.

Question (Beraisa): We bless on fire only on Motza'ei Shabbos, for that is when it was created.

R. Yehudah says, one says them (the Berachos on fire and fragrances) over a cup [of wine].

(R. Yochanan): The Halachah follows R. Yehudah. (He agrees that we bless only on Motza'ei Shabbos!)

Answer: On Motza'ei Yom Kipur one may bless on a fire that 'rested', but not on a fire that was initiated on Motza'ei Yom Kipur through sticks and stones.

(Beraisa #1): One may bless on a fire initiated through sticks and stones.

Contradiction (Beraisa #2): One may not bless on it.

Resolution: One may bless on it on Motza'ei Shabbos, but not on Motza'ei Yom Kipur.

On Motza'ei Shabbos Hash-m gave to Adam a semblance of Divine wisdom, and brought two rocks to him. Adam rubbed them together and fire came out.

Berachos 53a (Beraisa): If [on Motza'ei Shabbos] a Yisrael lit from a Nochri's flame, one may bless on it. If a Nochri lit from a Nochri, one may not bless on it.

We bless on the additional flame. Letter of the law, also when a Nochri lit from a Nochri, one may bless on it. We decreed not to, lest one bless on a Nochri's flame itself [that was used on Shabbos].

Rishonim

Rambam (Hilchos Shabbos 29:27): On Motza'ei Yom Kipur we bless only on a fire that rested.

Magid Mishneh: The Ramban (Berachos 50a DH v'Kivan) says that if a Yisrael lit from a Nochri, one may not bless on it on Motza'ei Yom Kipur, since one blesses on the increased Heter, and this is like a fire initiated now [through sticks and stones]. On Motza'ei Yom Kipur we bless only on a fire that rested or a fire lit from it. The custom in our lands is to bless on lanterns of the Beis ha'Keneses. It seems that the Rambam agrees.

Rashi (54a DH b'Ohr she'Shavas): A fire that rested is one that was burning today, but no Isur was done with it, for it was lit b'Heter, e.g. for a woman giving birth or for a Choleh, or it was burning from before Yom Kipur. We bless on it on Motza'ei Yom Kipur, for it is a Chidush to him. He did not benefit from it today.

Poskim

Shulchan Aruch (OC 624:4): [On Motza'ei Yom Kipur] we bless on fire, but not on fire created now from rocks.

Beis Yosef (DH v'Zeh, citing Kolbo 70, 36:3): We bless only on fire that rested from Melachah, e.g. it was burning the entire day, but not on a fire created [on Motza'ei Yom Kipur] through sticks and stones. The reason to bless on fire on [Motza'ei] Yom Kipur is because one ceased to benefit from it on Yom Kipur, and now it is permitted. Therefore, we require a fire that rested. Rest [on Yom Kipur] does not apply a fire created now. On Motza'ei Shabbos the entire year we are not particular about a fire that rested, since the reason we bless then is because its initial creation was on Motza'ei Shabbos. The Yerushalmi says that on Motza'ei Shabbos Hash-m gave to Adam two tiles. Adam banged them together and fire came out, and he blessed Borei Me'orei ha'Esh. Therefore, on Motza'ei Shabbos we bless on any fire, even if it came from sticks and stones, for this is like its creation. This reason does not apply on Yom Kipur, therefore we require a fire that rested.

Mishnah Berurah (7): Yom Kipur is unlike Shabbos. We bless on fire on Motza'ei Yom Kipur to show that it is unlike other Yomim Tovim, and it was forbidden to burn, and now it is permitted. The fire is a Chidush, and we bless on it. If Yom Kipur was on Shabbos, letter of the law one may bless on Motza'ei Yom Kipur on a fire initiated now [through sticks and stones]. The custom is to be stringent. In any case, surely one may be lenient about a fire lit from the fire from sticks and stones.

Kaf ha'Chayim (13): The Ra'avad says that one must pursue a fire that rested. The Ritva and his Rebbeyim say that one need not pursue it.

Shulchan Aruch (ibid.): Some permit blessing on it from the first pillar and onwards.

Source: Beis Yosef (ibid., citing Kolbo).

Magen Avraham (3): The first pillar is what was lit from the flame [that came from sticks and stones]. See Sa'if 5.

Gra (DH v'Yesh): This is a different fire, like it says in Berachos 53a. See the Magen Avraham (4, below).

Mishnah Berurah (9): Also this is called a fire that rested. It was not created new now.

Bi'ur Halachah (DH v'Yesh): Some Acharonim permit relying on this opinion in pressed circumstances, to bless on a Ner lit from a fire initiated now, but I did not bring it. The Magen Avraham proves from Sa'if 5 that one may not bless on the second Ner lit from a fire that did not rest. Shulchan Aruch ha'Rav and the Gra agree. Sa'if 5 is based on the Ramban, and the Magid Mishneh, Ritva, R. Yonah, Ro'oh, and Maharam Chalavah all brought it from him or in their own name. The lenient opinion in this Sa'if is a lone opinion. The Rema below is lenient to bless on a Ner of the Beis ha'Keneses together with a Ner lit from it, even though we do not bless on the former by itself, since it is for honor [and not for illumination]. This is unlike a fire initiated now. Seemingly, this shows that the Rema holds like the lenient opinion here. Really, this is wrong. Many Rishonim permit blessing on Motza'ei Yom Kipur on a Ner Beis ha'Keneses, for it is not for honor. In any case we may be lenient to join it with another Ner. The Rema could agree that one may not join a Ner with a fire initiated now.

Kaf ha'Chayim (17): In pressed circumstances, if one cannot get a fire that rested, one may rely on those who permit blessing on a fire initiated now, from the first pillar and onwards, rather than to omit blessing.

Shulchan Aruch (5): If a Yisrael lit a Ner from a Nochri's Ner, one may not bless on it on Motza'ei Yom Kipur, for we require a fire that rested during the day, or a Ner lit from it. The custom is to bless on lanterns of the Beis ha'Keneses.

Magen Avraham (4): The Magid Mishneh forbids blessing on a Ner lit from a fire initiated now. Since one may not bless on the primary flame, also on the addition one may not bless, for it did not rest during the day. R. Yonah says that some permit blessing on a flame lit from coals, and some disagree, for it is like what came from sticks and stones. This shows that all forbid blessing on a flame lit from what came from sticks and stones. Why did the Beis Yosef write in Sa'if 4 that some permit, and here he wrote Stam like the Magid Mishneh?

Gra (DH Yisrael): See the Magen Avraham. The Mechaber contradicts what he wrote in Sa'if 4.

Kaf ha'Chayim (18): In pressed circumstances one may rely on Teshuvas R. Menachem Azaryah (18), who permits blessing on what a Yisrael lit from a Nochri's Ner (Eliyahu Rabah 4).

Rema: Some say to say Havdalah on a Ner of the Beis ha'Keneses. Some say not to.

Taz (4): They hold that since it was only to honor the day, it is as if it did not burn at all.

Rema: Rather, we light a Ner from it. It is proper to say Havdalah over both together.

Mishnah Berurah (13): Also if one had a Ner in his house from before Yom Kipur, he lights another Ner from it and blesses on both together. Perhaps the former was to honor the day. Derech ha'Chayim says to leave a Ner burning from before Yom Kipur in order to bless on it on Motza'ei Yom Kipur, and fulfill all the opinions.

Mishnah Berurah (14): If one blessed only on the Ner Beis ha'Keneses, he need not bless again, since the Ner is also for light to enable prayer.