The elevator is operated by a lower-caste worker. In the era the novel was written, elevators were really simple machines that had to be operated by a worker, but in the science fiction future that Huxley sets the novel in, that job certainly could have been eliminated because they obviously have the technical expertise. So the “liftman” in that passage is literally performing the function of a machine, which makes him robot-like.
Furthermore, the elevator is, to some extent, automated: when the liftman delays taking the elevator back down after the passengers get off onto the roof (he delays because he’s so delighted to see the sun), an automated message kicks in that instructs him to take the elevator down to the next floor it’s needed on. So the lower caste liftman is also robot-like because he must slavishly obey the orders he receives.
In this scene, the elevator (or lift) carries Bernard and Lenina to a rooftop destination.
The "liftman" we encounter is an Epsilon-minus-semi-moron, part of the lowest caste in this society. He is robotic because he doesn't think, but simply responds to conditioning as he has been taught to do. As the elevator arrives at the rooftop, he cries "Oh, roof" in a voice of "rapture," then "Roof!" and later "roof?" in a way that makes us think of a dog's bark. He is described as "suddenly and joyfully awakened from a dark annihilating stupor."
Highlighting his animal or robotlike response, we learn that he:
smiled up with a kind of doggily expectant adoration into the faces of his passengers. Talking and laughing together, they stepped out into the light. The liftman looked after them.
He does not know what to do until he gets instructions from a recorded voice and then he obeys, falling back into his "habitual stupor," like a robot turned off. This vignette is played for laughs, but it reflects a dark humor, showing a human being designed for the convenience of others to have no more intelligence than an animal or a machine.