Why did Rav Huna need a verse to permit lighting the fire in Beis ha'Mokad on Erev Shabbos?

Rav Elyashiv: If not for the verse, we would decree lest they stoke. Even though Ein Shevus b'Mikdash, not all Shevusim (mid'Rabanan laws) were permitted, like Tosfos said (123b). The verse teaches that there is no Isur to burn in the Mikdash. This is fine according to the Rambam, that the fire of Beis ha'Mokad is used for the Mizbe'ach. However, Rashi holds that it is to warm Kohanim. This is not needed for Avodah! Even so, since it is in the Mikdash, which is not Moshvoseichem, the verse excludes (permits) it. (NOTE: Rav Elyashiv needed to prove that not all Shevusim are permitted according to the Rambam. According to Rashi, Esh Beis ha'Mokad is not needed for Avodah, so Ein Shevus b'Mikdash does not apply to it! - PF)

In what case does the verse permit burning limbs and Chelev?

Rashi: It permits the entire night. However, the fire of Beis ha'Mokad is not needed for Avodah. It is for Kohanim - it is not permitted.

Tosfos: This is wrong. We expound below (24b) "Olas Shabas b'Shabato" - but one may not burn Olas Chol (of Erev Shabbos) on Shabbos!

Tosfos: It does not permit burning Olah of Shabbos on Shabbos, even though one may burn them the entire night, after Shabbos. We learn this from "Olas Shabas b'Shabato." Since it excludes Olas Chol, for which Shechitah and Zerikah are during the day (Erev Shabbos), it must permit limbs and Chelev of Olas Chol, if they caught fire during the day. We expound to forbid Olas Chol on Shabbos if it did not catch fire during the day.

Daf Al ha'Daf citing Ein Yitzchak (1 EH 80, from the author's son): Pnei Yehoshua (Berachos 20) says that when the Aseh [that overrides a Lav] is written by the Lav, there is no Isur at all. It is not Nidcheh, rather, Hutrah. E.g. Tum'as Nazir is Hutrah for a Mes Mitzvah, and Tum'as Kohen is Hutrah for close relatives. If so, also burning on Shabbos is Hutrah, for we say that "Lo Seva'aru Esh b'Chol Moshvoseichem" permits burning limbs and Chelev. This is unlike his father wrote in Be'er Yitzchak (OC 14:3) that limbs and Chelev are burned on Shabbos only because the Mitzvah is dear in its time! Also the Rambam wrote that Tum'as Kerovim is Nidcheh (unlike the Pnei Yehoshua).

What is the answer 'Kohanim are zealous'?

Rashi: They are all Bnei Torah and zealous [to guard Torah]. They will not come to stoke the fire after dark.

Daf Al ha'Daf: The Kesef Mishneh (Hilchos Beis ha'Bechirah 8:12) says that Ein Shevus b'Mikdash because Kohanim are zealous. The Sefas Emes (Beitzah 20a) asked that we may put Pesach to roast close to Shabbos, because a Shevus of the Mikdash, also outside the Mikdash they did not decree. Abaye holds that people of the group [who will eat Pesach] are not zealous. This shows that Ein Shevus b'Mikdash is not because Kohanim are zealous! Sho'el u'Meshiv (1:2:146) says that Ein Shevus b'Mikdash applies when there is concern for a Torah Isur, but not when Chachamim forbade because it looks like a Torah Melachah. Even if Kohanim are zealous, this Melachah is forbidden mid'Rabanan! This resolves Rashi (Menachos 97), who says that even though Ein Shevus b'Mikdash, moving [sticks that separate loaves of Lechem ha'Panim] is not due to Shevus. Rather, it looks like building and destroying. Even if Kohanim are zealous, we are concerned lest people permit building and destroying elsewhere.

Rav Elyashiv: Even if we find Kohanim who are not Bnei Torah and zealous, Chachamim did not distinguish; they did not decree.