The robots saying "Eliminate!" is a reference to the Daleks in Doctor Who saying "Exterminate!."
The robot who saves Homer's life by pushing him out of the path of an oncoming truck (and being himself hit as a result) is similar to the self-sacrificing robot in the 100th episode of The Twilight Zone, "I Sing the Body Electric."
When Smithers is throwing books at the robots he says "to servos with love," a reference to the movie To Sir with Love.
The jersey on one robot says "Nomad J25," which may be a reference to the main character in the 1956 novel The Stars My Destination.
Lenny playing the "world's smallest violin" is a possible reference to the SpongeBob SquarePants episode: "Squilliam Returns" or the movie Reservoir Dogs (1992) quote of Mr. Pink.
When Barney becomes a stripper he says that The Full Monty gave him the inspiration.
The summary of the episode is a reference to the Mexican animated film, Top Cat: The Movie, as they both involve robots taking over almost everyone's jobs.
The couch gag of this episode shows a time lapse shot (spanning 1989-2012) of the Simpson family sitting in front of a banner reading "America, Most powerful country in the world." The banner slowly degrades over time as characters from past episodes appear. Things and characters depicted include:
The last time robots worked at the Springfield Power Plant while Homer still worked there was in "Lisa's Wedding." Or rather, this is the first time, and that is the second time, given the idea that that is set in the future.
Homer leaves Mr. Burns's office, then a few seconds later, Burns turns around to the CCTV monitor and sees Homer in Sector 7-G putting his stuff back on his desk.
Lisa says the robots can only run three mph, though later in the episode, they run about as fast as the hounds. The hounds could easily run at three mph.
In this episode it was revealed that Mr. Burns was once nice while in adulthood. However, in "Rosebud" he turned evil as a child. He could have repented and then become the heartless villain he is now. Not clear if he was nice. The flashback involves Burns (who has just opened the Springfield Power Plant) organizing a parade and trying to win over the locals. It seems to have been a public relations campaign.