You certainly don't have to peel them, although if you don't then you'll want to have a food mill and a certain amount of time available (just 'straining' ain't gonna remove the peels and seeds, you need a food mill or similar)
However, remember that if you do that, whatever contaminants are on the peels will end up in the applebutter that way, and in concentrated form to boot.
Personally I would not do it with typical storeboughten non-organic apples (especially if they are from Chile or such). (And I am not generally especially paranoid about such things). For borderline cases, you can remove the bulk of any pesticide-residue contamination by removing the area right around the stem and flower ends of the apple; I do that sometimes for applesauce, just use a paring knife to slice out a shallow cone of peel from around each "pole" of the apple, then quarter the apple and chuck it into the pot with core and most of the peel still on. Then run thru the Foley food mill once things are all mush.
Good luck, have fun,
Pat