"BURN-E" stands for Basic Utility Repair Nano Engineer.
(at around 1 min) The song BURN-E sings to himself while starting the spire repairs is Ludwig van Beethoven's 9th symphony, 'Ode to Joy'.
(at around 6 mins) The planetary zoom-out at the end of the short shows the Axiom landing site to be in the vicinity of Richmond, Virginia.
The lift to take BURN-E outside is labeled LV-426, which is the designation for the planet in Aliens (1986).