I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain topics is objectionable (especially in the foundational chapters II and III). Here I'm not talking about the mechanics of the presentation -- being too terse, leaving important theorems to the exercises, and so on. Rather, I take these comments to mean that he develops certain parts of the theory in ways that are philosophically questionable, or technically "hacky." In particular, his treatment of sheaves and cohomology is often singled out, though this is far from the only thing I've heard people grumble about.
What parts of the theory does Hartshorne do in a way that might be considered morally "wrong," and what would a "correct" treatment look like?