Electronic components suffer from infant mortality. If they are going to break, they will break early, or last a long time. 'Burning in' is a technique where a component (or complete product) is run at elevated temperature to encourage devices that would fail early in life to pack up immediately. This is used where warranty claims need to be managed or in 'mission critical' devices (jet engine controls, medical equipment etc etc) where failure has consequences. This has nothing to do with HiFi.
'Running in' is where a mechanical system changes in performance over time as mating surfaces wear together, or where the characteristics af a component change with use. This could apply to the mechanical components in a speaker - the compliance of the cone surround and voicecoil spider MAY change with use, and this MAY impact the speaker performance.
Applying the term 'burning in' to amplifier performance smells of snake oil.