That's what unquenchable means by very definition.
"unquenchable (adj.) late 14c., of fire; 1560s, of thirst, from un- (1) "not" + quench (v.) + -able. Related: Unquenchably.unquenchable (adj.) late 14c., of fire; 1560s, of thirst, from un- (1) "not" + quench (v.) + -able. Related: Unquenchably." -
https://www.etymonline.com/word/unquenchable
"quench (v.) Middle English quenchen, "to extinguish, put out" (heat, light, fire, also of desire, hunger, thirst), also figurative, "to bring to naught, eliminate, render ineffectual" (c. 1200), Old English acwencan "to quench" (of fire, light), from Proto-Germanic *kwenkjanan, probably a causative form from the source of Old English cwincan "to go out, be extinguished," Old Frisian kwinka. No certain cognates outside Germanic; perhaps a substratum word. Especially "to cool or extinguish by means of cold water," hence "to drench in water" (late 15c.). Related: Quenched; quenching." -
https://www.etymonline.com/word/quench?ref=etymonline_crossreference#etymonline_v_3177
Just in case you do not understand "extinguish"
"extinguish (v.) "to put out, quench, stifle," 1540s, from Latin extinguere/exstinguere "quench, put out (what is burning); wipe out, obliterate," from ex "out" (see ex-) + stinguere "quench," apparently an evolved sense from PIE *steig- "to prick, stick, pierce" (see stick (v.)). But see distinguish (v.). Related: Extinguished; extinguishing.extinguish (v.) "to put out, quench, stifle," 1540s, from Latin extinguere/exstinguere "quench, put out (what is burning); wipe out, obliterate," from ex "out" (see ex-) + stinguere "quench," apparently an evolved sense from PIE *steig- "to prick, stick, pierce" (see stick (v.)). But see distinguish (v.). Related: Extinguished; extinguishing." -
https://www.etymonline.com/word/extinguish#etymonline_v_14119
Isaiah speaking about Jerusalem when Nebuchadnezzar would later come:
Isa 34:10: "It shall not be quenched night nor day; the smoke thereof shall go up for ever: from generation to generation it shall lie waste; none shall pass through it for ever and ever."
Jerusalem burned so hot by fire that no one could put it out by any means. Jerusalem burned up with unquenchable fire, and once the fuel ran out, nothing of the fire remained, and only ashes left.
In the contexts of the NT texts referring to the finally impenitent, they are burned "up" by unquenchable fire. They cannot deliver themselves from the power of the flame, and when they are consumed away, the fire ceases for lack of fuel, but no one quenched it.