why not "he has to move it for me"?
That has a completely different meaning, it would be: הוא צריך להעביר אותו(/ את זה) בשבילי
You are right that "להעביר" can be translated as "to move" too, but in this context the only acceptable translation is the one above.
Ah. I see now. Thank you!
What's wrong with "He needs to pass me him"? (e.g. a baby)
The same question. Natives please explain.
Well, if the pronoun refers back to the rare case that you get passed a person like a תִּינוֹק, your translation should be correct. I think you can use it also to say, that someone should put someone other on the phone.
why is it אותו and not זה? he needs to pass it (את זה)
It's אותו because a masculine object is implied. It could be anything from מלח or אוכל or maybe מזלג. On the other hand את זה would be used if you talk about something in general or in theory.