= Solution
The point to retain in the direct-image theorem is <quasi-compactness> of inverse images of affine opens and of overlaps; no separation assumption has been supplied. The underlying space of a <Noetherian scheme> is Noetherian, so every <open subset> of $X$ is <quasi-compact>. Fix an <affine open subscheme> $V=\operatorname{Spec}A$ of $Y$, put $W=f^{-1}V$, and choose a finite <open cover> by <affine open subschemes> $W=\bigcup_i U_i$. For each pair $i,j$, choose a finite affine cover $(W_{ij\ell})_\ell$ of $U_i\cap U_j$. Empty overlaps contribute no terms.
Write $M=\Gamma(W,\mathcal F)$. The <sheaf gluing axiom> gives the <exact sequence>
$$
0\longrightarrow M\longrightarrow
\prod_i\Gamma(U_i,\mathcal F)
\xrightarrow{\delta}
\prod_{i,j,\ell}\Gamma(W_{ij\ell},\mathcal F),
$$
where $\delta$ is the difference of the two restrictions to each overlap chart. All terms are $A$-<modules> through $f$. For $a\in A$, the inverse image of $D(a)$ cuts each $U_i$ and $W_{ij\ell}$ by a <principal open subscheme>. Because $\mathcal F$ is a <quasi-coherent sheaf>, sections on these smaller affine charts are the corresponding <module localizations> at $a$. <Exactness of localization> and its commutation with finite products now identify the localized equalizer with the equalizer for the restricted cover. Thus
$$
\Gamma(f^{-1}D(a),\mathcal F)\cong M_a.
$$
The isomorphisms respect restrictions, proving $(f_*\mathcal F)|_V\cong\widetilde M$ on the basis of <principal open subschemes>. Since $V$ was arbitrary, \b[$f_*\mathcal F$ is a quasi-coherent sheaf]. The finite covers of overlaps are what allow the proof to work for nonseparated <Noetherian schemes>; this is the <quasi-coherence of direct image under a quasi-compact quasi-separated morphism>.
For a <quasi-coherent sheaf> which is not coherent but has coherent <direct image>, use $f:\mathbb P^1_k\to\operatorname{Spec}k$ and
$$
\mathcal F=\bigoplus_{n=1}^{\infty}\mathcal O_{\mathbb P^1}(-1).
$$
This <infinite negative-twist sum with zero global sections> is quasi-coherent: on each standard <affine chart> it is the sheaf associated with a direct sum of free rank-one <modules>. Its <stalk> at every point, modulo the <maximal ideal>, is an infinite-dimensional <vector space>. A finitely generated <module> would have a finite-dimensional quotient, so $\mathcal F$ is not a <coherent sheaf>.
Nevertheless, $\Gamma(\mathbb P^1,\mathcal O(-1))=0$. This follows from <cohomology of twisting sheaves on projective space>, or directly by gluing on the two standard affine charts: if $z=t_1/t_0$, a section is a polynomial $p(z)$ on the first chart and $q(z^{-1})$ on the second with $p(z)=z^{-1}q(z^{-1})$, which forces both to vanish. <Global sections> commute with this direct sum: they are the kernel of the difference map for the two-chart cover, and <direct sums> commute with that finite equalizer of <modules>. Hence
$$
\boxed{f_*\mathcal F=0,}
$$
which is coherent on $\operatorname{Spec}k$. Both <schemes> in this example are Noetherian.
For a <finite morphism>, such an example is impossible. On an <affine open subscheme> $V=\operatorname{Spec}A$ of the target, its inverse image is $\operatorname{Spec}B$, with $B$ a finite $A$-<module>. Write $\mathcal F|_{f^{-1}V}=\widetilde M$. Its <direct image> corresponds to $M$ considered as an $A$-<module>. If the <direct image> is coherent, $M$ is finitely generated over $A$. The same generators also generate it over $B$, because $A$ acts through $B$. Since $B$ is Noetherian, $\widetilde M$ is coherent. Conversely, a finite set of $B$-generators combined with a finite set of $A$-generators of $B$ gives finitely many $A$-generators of $M$. Thus <coherence reflected by finite direct image> gives the stronger equivalence
$$
\boxed{\mathcal F\text{ coherent}\iff f_*\mathcal F\text{ coherent}
\quad(f\text{ finite}).}
$$
Finally, let $Y=\operatorname{Spec}k[x,y]$, let $X=Y\setminus\{(x,y)\}$ be the <punctured affine plane>, and let $j:X\hookrightarrow Y$ be the <open immersion>. Both are <integral schemes>, but $j$ is not an <isomorphism of schemes> because it omits a point. The cover $X=D(x)\cup D(y)$ gives
$$
\Gamma(X,\mathcal O_X)
=k[x,y]_x\cap k[x,y]_y=k[x,y]
$$
inside the <field of fractions> $k(x,y)$. Indeed, in a reduced fraction, membership in the first <localization> forces every denominator factor to be associated to $x$, while membership in the second forces it to be associated to $y$. <Unique factorization> and coprimality force the denominator to be a <unit>. By the theorem just proved, $j_*\mathcal O_X$ is quasi-coherent on the affine $Y$, hence determined by this <module> of <global sections>. The natural map $\mathcal O_Y\to j_*\mathcal O_X$ corresponds to the identity of $k[x,y]$, so
$$
\boxed{j_*\mathcal O_X\cong\mathcal O_Y,}
$$
a <coherent sheaf>. This is a concrete case of <codimension-two extension of regular functions on a normal variety>.
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