Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

The short answer is that quasi-compactness is enough (for the statement you asked about): see lemma 3.1 in http://www.math.columbia.edu/algebraic_geometry/stacks-git/coherent.pdf (but its essentially the argument in Hartshorne)

this business of extra hypothesis comes up because Hartshorne and Gronthendieck are proving an iff statement; that is, these hypothesis are needed to prove that if $X$ is affine then $H^1(X,F) = 0$ for every quasicoherent sheaf. In the case of Hartshorne you need notherian hypothesis to prove

lemma II.3.3 If $I$ is an injective $A$-module and $f \in A$ then $I \to I_f$ is surjective.

with this lemma you can then show if $I$ is injective then $\tilde I$ is flasque an so you can use them to calculate cohomology: $F$ is quasicoherent $\Rightarrow F = \widetilde{M}$. If $M \to I_0 \to I_1 \to ...$ (1) is an injective resolution then $\widetilde{M} \to \widetilde{I_0} \to \widetilde{I_1} \to ...$ is flasque resolution and applying global sections just recovers (1) so all the higher cohomology vanishes.

I'm not familiar with Gronthendieck's argument and I don't know if you can replace separated with quasi-separated.