Chapters 2 and 3 comprise the majority of the five chapters in Harshorne's text -- something like $233$ out of $423$ pages -- and they treat schemes in much more generality than just varieties over an algebraically closed field. So taking out the schemes would remove more than half of the text!
[As an aside: the first algebraic geometry class I took -- in my first semester of grad school -- used Hartshorne's text but did not discuss scheme theory. Trying to read Hartshorne and restricting to varieties instead of schemes was very confusing: imagine trying to read about "separated and proper morphisms" when you only know about varieties!]
If you mean to ask why does he assume the ground field is algebraically closed in Chapters 4 and 5 when much of what he does is true without that hypothesis: well, that's a choice that he made, and no one but he could definitively answer it. As an arithmetic geometer I certainly wish that he had treated more general ground fields. To a large degree Qing Liu's recent text Algebraic geometry and arithmetic curves exists because of this restriction in Hartshorne's text. (More precisely it exists because Hartshorne did not treat arithmetic surfaces.) Still Hartshorne does more with cohomology of schemes than Liu does, so a student of arithmetic geometry should spend time reading both texts.