[When WALL-E tries to shield EVE from the sandstorm she pulls her gun him.]
EVE isn't exactly warm and cuddly. If she had a knowledge of reality television, she'd be the type to say, "I'm not here to make friends," because she's not on Earth to make friends; she's programmed to find plant life.
[EVE scans the plant and shuts down, a green light blinking on her chest.]
Here we realize that EVE's duty is to safeguard the plant. She puts it inside her body and shuts down. We doubt anything could get inside her impenetrable shell—hmm, that's a handy metaphor for her emotions, too. Duty first, love second.
COMPUTER: Captain, you are needed on the bridge.
The word "needed" isn't quite right, here. The Captain isn't "needed" to anything. He's a figurehead, although much more plump and less beautiful than the figureheads we're used to.
CAPTAIN: Honestly, it's the one thing I get to do on this ship.
Even though the Captain's job is mostly useless, like Sigourney Weaver's in Galaxy Quest(appropriate, since in WALL-E she plays an actual computer) but he takes pride in his one duty: morning announcements. At least, he takes pride in it when he gets up early enough to do it.
[M-O leaves the green line to clean WALL-E's tracks.]
M-O's duty is to travel a green line and clean anything near it, but WALL-E's tracks are so dirty he abandons his post to follow them. Cleaning WALL-E becomes his new obsession.
AXIOM PASSENGERS: Yay, Captain! Captain, we're home, it's so beautiful.
CAPTAIN: No, it's nothing. I was pleased to do this. It's all about you people. It's not about me.
The Captain is fantasizing about doing something more than just sitting in his Captain's chair all day. Something little, like, oh, saving humanity. All in a day's work for our intrepid Captain (in his imagination).
CAPTAIN [to the plant]: Just needed someone to look after you, that's all…
When EVE returns with the plant, the Captain waters it, and he talks to it. He seems to apply this same sense of duty to Earth itself, like he needs to go look after it (conveniently forgetting that humans screwed everything up in the first place!)
AUTO: Sir, I insist you give me the plant. […] We cannot go home.
AUTO is just following orders that were programmed into him. Does that mean he's evil, or just (unlike everyone else in this film) just following his job description?
CAPTAIN: Tell me what's classified. […] Tell me, AUTO! That's an order.
Here's a little robot conundrum. It seems that AUTO both has to follow his programming directive and orders from the Captain. Why doesn't the Captain just order AUTO to send them back to Earth?
BNL PRESIDENT [in a recording]: Operation Cleanup has, well, uh, failed. Wouldn't you know, rising toxicity levels have made life unsustainable on Earth. […] Darn it all, we're going to have to cancel Operation recolonize. So just stay the course. Rather than try and fix this problem it'll just be easier for everyone to remain in space."
Here we get the full scope of AUTO's duty. It seems that humans gave up on Earth 700 years ago, and decided that they should just stay in space instead of trying to fix things.