Yep, this is one of those sentences where one can only hope the producers of this program were having fun. And I see no "it" here, either...
there is definitely missing some context. When are we supposed to use smth like that?
Maybe they're talking about how well a robot works, and they refer to him as "he."
Or maybe the subject is pulling a corpse to their car, pushing it in the trunk and making sure it's locked; we'll never know :P
It certainly sounds like he's dealing with a sticky door, but I see no clitic pronoun. Duo accepted "He pulls, pushes, and closes well" although I'm thinking this really means "He pulls and pushes, and it closes tight."
Duo gave me the sentence a second time, and it does accept "He pulls, pushes, and it closes well." I didn't take a risk on "tight."
For Duo sentences, I either imagine a children's book with big pictures and one sentence per page (e.g. "The bear eats a fish") or a murder mystery (e.g. this sentence, where the detective is obviously sneaking into someone's house.) :-)
Objects have gender, therefore you would refer to "it" as "lui" or "lei"
"Literary subject pronouns also have a distinction between animate (egli, ella) and inanimate (esso, essa) antecedents, although this is lost in colloquial usage, where lui, lei and loro are used for animate subjects as well as objects, while no specific pronoun is employed for inanimate subjects."
If the need arose, you would drop "lui". Of course, that could still be translated as "he" or "she", but as far as I know, there is no word for just "it" in common use today.