…do not extinguish it. (Leviticus 6:6)
The same verse that requires the altar to be equipped with an eternal flame likewise prohibits extinguishing same. The Talmud in Yoma (46a) tells us that one is liable for extinguishing this fire even if he removed it from the altar before doing so. However, if fire were taken from the altar for another, permitted purpose (such as to light the Menorah) and someone extinguished it, he would not be liable since it was no longer considered altar fire at that point.
The underlying rationales for this mitzvah are the same as for the previous one: as a source of blessing to the force of nature within man and to minimize the appearance of miracles.
This prohibition applies to both men and women in Temple times. It is discussed in the Talmudic tractate of Zevachim on page 91b. It is codified in the Mishneh Torah in the second chapter of Hilchos Tamidin and is #81 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.