Gemara

Beraisa - R. Yehudah: Every (other) day there are two Ma'arachos on the Mizbe'ach. Today (Yom Kipur) there are three:

There is a big Ma'arachah (for burning Korbanos), a small Ma'arachah from which we take coals to burn the daily Ketores, and an additional Ma'arachah on Yom Kipur.

R. Yosi says, every day there are three. Today there are four:

There is a Ma'arachah for Kiyum ha'Esh, in addition to the above two or three.

R. Meir says, every day there are four. Today there are five:

There is a Ma'arachah for burning limbs and Chelev left from yesterday, in addition to the above three or four.

Question: All agree that there are at least two Ma'arachos. What is the source?

Answer: "Hi ha'Olah Al Mokdah Al ha'Mizbe'ach Kol ha'Laylah" is the big Ma'arachah. "V'Esh ha'Mizbe'ach Tukad Bo" is the Ma'arachah for Ketores;

R. Yosi learns about Kiyum ha'Esh from "Veha'Esh Al ha'Mizbe'ach Tukad Bo";

R. Yehudah says that this teaches about lighting the big Ma'arachah:

(Beraisa - R. Yehudah) Question: What is the source that the big Ma'arachah must be lit on top of the Mizbe'ach (i.e. we may not bring up burning wood)?

Answer: We learn from "Veha'Esh Al ha'Mizbe'ach Tukad Bo".

Question: R. Yosi uses this verse to teach about for Kiyum ha'Esh. What is his source for R. Yehudah's law?

Answer: He learns like R. Shimon:

Beraisa - R. Yehudah: "V'Nosnu Benei Aharon ha'Kohen Esh Al ha'Mizbe'ach" teaches that a Kosher Kohen must light the big Ma'arachah, with Klei Shares (Bigdei Kehunah - Rashi).

Objection (R. Shimon): We already know that a Zar may not approach the Mizbe'ach!

R. Shimon: Rather, it teaches that the big Ma'arachah must be lit on top of the Mizbe'ach.

R. Yehudah: The verse is needed to forbid a Zar to light the Ma'arachah using a bellows while standing on the ground.

Question: What is R. Meir's source for a Ma'arachah to burn limbs and Chelev left from yesterday?

Answer: He learns from "V'Esh".

R. Yosi and R. Yehudah do not expound the 'Vov'. They hold that leftover limbs and Chelev are returned to the big Ma'arachah:

Beraisa - Question: Limbs and Chelev left from yesterday are arranged on the Mizbe'ach. If there is no room, they are arranged on the ramp or Sovev until the big Ma'arachah is made. What is the source for this?

Answer: "Asher Tochal ha'Esh Es ha'Olah Al ha'Mizbe'ach".

R. Meir says that this teaches that we return limbs of an Olah that flew off the Mizbe'ach, but we do not return Ketores.

Question: All agree that there is an extra Ma'arachah on Yom Kipur. What is the source?

Answer: They learn from "Veha'Esh". Even the opinion that does not expound a 'Vov' agrees that we expound 'veha'.

Zevachim 91b - Shmuel: A Nedavah of wine is sprinkled on the fire (on the Mizbe'ach).

Question: The wine will (partially) extinguish the fire (where it lands). This is forbidden ("Esh Tamid Tukad Al ha'Mizbe'ach Lo Sichbeh")!

Answer #1: It is forbidden only to totally extinguish the fire.

Question: But Rav Nachman taught that one who takes a coal off the Mizbe'ach and extinguishes it is liable!

Answer: Rav Nachman discusses when no fire was left on the Mizbe'ach.

Answer #2: (Normally, one is liable even if some fire remains on the Mizbe'ach.) Extinguishing is permitted for a Mitzvah (e.g. a Nedavah of wine).

Question (Beraisa - R. Eliezer ben Yakov) Suggestion: Since Terumas ha'Deshen is a Mitzvah, perhaps one may extinguish glowing ashes for this!

Rejection: "Lo Sichbeh."

Answer: There it is forbidden, for there is no need to extinguish them. One can wait for them to extinguish by themselves.

Rishonim

Rambam (Hilchos Temidim u'Musafim 2:4): Every day there are three Ma'arachos on the Mizbe'ach. The first is the big Ma'arachah for burning the Tamid and other Korbanos. Next to it is a small Ma'arachah from which we take coals in a pan to burn the daily Ketores. Nothing is burned on the third Ma'arachah. It is to fulfill the Mitzvah of having fire on the Mizbe'ach, "Esh Tamid Tukad..."

Lechem Mishneh: Rashi explains 'Kiyum ha'Esh' to mean to sustain the fire on the big Ma'arachah in case it dwindles The Rambam explains it to mean fulfilling the Mitzvah of having fire on the Mizbe'ach.

Si'ach Yitzchak (45b DH Tamid): The Rambam holds that the fire in Beis ha'Mokad was used to relight the big Ma'arachah if necessary. The Ma'arachah for Kiyum ha'Esh was solely for the Mitzvah. Alternatively, it could be used to light the Menorah of the big Ma'arachah. However, it was not needed on a regular basis.

Mizrachi (and Gur Aryeh Vayikra 6:5 DH veha'Esh): Two logs are placed each morning for the Ma'arachah for Kiyum ha'Esh.

Rebuttal (Lechem Mishneh): The Ma'arachah for Kiyum ha'Esh is in addition to the two logs. The two logs are learned from "U'Vi'er Aleha ha'Kohen Etzim ba'Boker", and this Ma'arachah is learned from "Veha'Esh Al ha'Mizbe'ach Tukad Bo". The Rambam (6:1) explicitly says that the two logs are placed on the big Ma'arachah.

Question: When the Rambam lists the daily Avodos (6:1), he does not mention the Ma'arachah for Kiyum ha'Esh!

Answer #1 (Radvaz 6:1): That Ma'arachah was always burning. The Rambam did not need to say that when it dwindles, the Kohen who merited to arrange the other Ma'arachos would add wood for Kiyum ha'Esh, for this is obvious.

Answer #2 (Lechem Mishneh 2:4): He omits it because there was not a fixed time for it.

Rambam (Hilchos Avodas Yom Kipur 2:5): Every day there are three Ma'arachos on the Mizbe'ach. Today we add a fourth, to beautify and glorify the Mizbe'ach.

Likutei Halachos (45b Ein Mishpat 2): We learn the extra Ma'arachah of Yom Kipur from "Veha'Esh". This implies that it is a proper Mitzvah mid'Oraisa; we Mechalel Yom Kipur for it! Rashi (DH she'Mosifin) says that from it we take coals for the Ketores burned in the Kodesh ha'Kodoshim. Perhaps the Rambam gives the reason for the Mitzvah (why we do not take from the usual Ma'arachah for Ketores).

Acharonim

Minchas Chinuch (132 DH ul'Aniyas): Surely, we fulfill the Mitzvah of constantly having fire on the Mizbe'ach through any Ma'arachah. R. Yehudah agrees that it is a Mitzvah to constantly having fire, nevertheless he does not require a separate Ma'arachah for this.

Question (Griz Zevachim 91b DH): R. Eliezer ben Yakov held that Terumas ha'Deshen would override Kiyum ha'Esh, if not for the Lav "Lo Sichbeh". If no other fire remains, surely it does not (override another Aseh)! If other fire remains, surely it is permitted!

Note: The Gemara did not resolve whether or not it is forbidden when other fire remains. It was unable to prove this from R. Eliezer ben Yakov.

Answer (Griz ibid.): We must say that fire of Shomayim remains. The conclusion is that extinguishing the man-made fire is considered totally extinguishing, even though fire of Shomayim remains.

Note: If without "Lo Sichbeh" R. Eliezer would permit to extinguish because Kiyum ha'Esh is fulfilled through Esh Shomayim, how does he learn from Lo Sichbeh otherwise? We should say that it merely adds a Lav when there is no Esh Shomayim! Perhaps it is unreasonable that a Lav for all generations should apply only when Esh Shomayim departs because of sins of Yisrael. However, it is reasonable that even though Kiyum ha'Esh is fulfilled through Esh Shomayim, it is a Mitzvah for people to bring fire as well.