Middle English flaume, also flaumbe, flambe, flame, flamme, mid-14c., "a flame;" late 14c., "a flaming mass, a fire; fire in general, fire as an element;" also figurative, in reference to the "heat" or "fire" of emotions, from Anglo-French flaume, flaumbe "a flame" (Old French flambe, 10c.), from Latin flammula "small flame," diminutive of flamma "flame, blazing fire," from PIE *bhleg- "to shine, flash," from root *bhel- (1) "to shine, flash, burn."
The meaning "a sweetheart, object of one's passion" is attested from 1640s; the figurative sense of "burning passion" was in Middle English, and the nouns in Old French and Latin also meant "fire of love, flame of passion," and, in Latin "beloved object." The Australian flame-tree is from 1857, so called for its red flowers.
Middle English flaumen, also flaumben, flomben, flamben, flamen, flammen, c. 1300 (implied in flaming "to shine (like fire), gleam, sparkle like flames;" mid-14c. as "emit flames, be afire, to blaze," from Anglo-French flaumer, flaumber (Old French flamber) "burn, be on fire, be alight" (intransitive), from flamme "a flame" (see flame (n.)).
Transitive meaning "to burn, set on fire" is from 1580s. Meaning "break out in violence of passion" is from 1540s; the sense of "unleash invective on a computer network" is from 1980s. Related: Flamed; flaming. To flame out, in reference to jet engines, is from 1950.