1540s, "having life, not dead," a shortening of alive (q.v.). From 1610s of fire, coal, etc., "burning, glowing;" 1640s of things, conditions, etc., "full of active power;" sense of "containing unspent energy or power" (live ammunition) is from 1799.
Meaning "in-person, not recorded" (of performance) is attested by 1917. Live wire is attested from 1890, "circuit through which an electric current is flowing;" figurative sense of "active person" is from 1903. Jocular real live "genuine" is from 1887. The older adjective is lively.
A GRIM RECORD — The death harvest of the "live wire" and "third rail" goes right on. It is not governed by seasons nor, qualified by time. It is the ubiquitous epidemic of electricity, defiant of doctors and ruthless as fate. [The Insurance Press, Aug. 22, 1900]