The Sacred Flame was fuelled by an underground natural gas pocket that escaped through a fissure in the rock, acting like a natural gas nozzle.
The Elixir was created from water from an underground spring. It passed through the rocks heated by the Sacred Flame directly below them. The rocks contained rare minerals and compounds that were released into the water when heated by the Flame.
Over countless millennia, carbon and soot had clogged the nozzle. The Flame had been reduced to barely more than a inch high, resulting in almost no generation of the Elixir. The Fourth Doctor solved the problem by placing a firecracker wrapped in blue touch paper in the fissure, blasting the soot loose and re-igniting the Flame. (TV: The Brain of Morbius)
When the Eighth Doctor died with a gunship pilot who refused to be rescued by a Time Lord from her crashing ship, the Sisterhood revived him long enough to regenerate for battle in the Last Great Time War. He quickly recognised that he had crashed on Karn and stood before the Sisterhood again. Annoyed by their presence, he remarked that they were the "Keepers of the Flame of Utter Boredom". Ohila did not appreciate his irreverent words and defended it as the Flame of "Eternal Life". (TV: The Night of the Doctor)