If $f:X\to Y$ is separated of finite type between qcqs noetherian schemes and $f_*$ preserves coherence, then $f$ is proper. A proof is as follows:
Since it is possible to extend a coherent sheaf on an open of $X$ to the whole of $X$ [Stacks project, Tag 01PD], we see that the statement is local on $Y$, and we are reduced to the case where $Y$ is the spectrum of a local ring.
By Nagata's compactification theorem, we may find a compactification of $f$: it is the composition of an open immersion with dense image $i:X\to \overline{X}$ and a proper morphism $\overline{f}:\overline{X}\to Y$. We suppose for contradiction that $Z=\overline{X}\setminus X$ is not empty.
Let us reduce to the case where $Z$ is finite and for every $z\in Z$, $\mathcal{O}_{\overline{X},z}$ is of dimension $1$. We do it by applying iteratively the following procedure. Choose a closed point $z\in Z$ at which $\mathcal{O}_{\overline{X},z}$ is not of dimension $1$. Using prime avoidance, choose a function $f\in \mathcal{O}_{\overline{X},z}$ that does not vanish on any irreducible component of $\overline{X}$ through $z$ nor on any positive-dimensional irreducible component of $Z$ through $z$. Replace $\overline{X}$, $X$ and $Z$ by their intersection with the closure of $\{f=0\}$ in $\overline{X}$. The hypothesis of preserving coherence still holds because closed immersions preserve coherence. The Hauptidealsatz ensures that $X$ is still dense in $\overline{X}$.
Using noetherian induction on $Z$, and induction on the dimensions of the local rings $(\mathcal{O}_{\overline{X},z})_{z\in Z}$, we see that the procedure terminates. Hence, from now on, we suppose that $Z$ is finite and for every $z\in Z$, $\mathcal{O}_{\overline{X},z}$ is of dimension $1$. Replacing moreover $\overline{X}$ by one of its components through $z$, we may assume that it is integral of generic point $\eta$.
Now we distinguish two cases. If $f(X)$ is the closed point of $Y$, then $X$ is an affine curve over the residue field of $Y$, and $f_*\mathcal{O}_X$ is not coherent: a contradiction.
Otherwise, consider the Stein factorization $\overline{X}\to S\to Y$ of $\overline{f}$. Up to replacing $Y$ by $S$ (we do not lose the hypothesis of preserving coherence, because $S\to Y$ is finite), we may suppose that $\overline{f}$ satisfies $\overline{f}_*\mathcal{O}=\mathcal{O}$. In particular, it is surjective with connected fibers.
The points of $Z$ are closed points, hence sent to the closed point of $Y$. Since their only generization in $\overline{X}$ is $\eta$, and since the fibers of $\overline{f}$ are connected, each of them is a fiber of $\overline{f}$. It follows that $X$ is the pull-back by $\overline{f}$ of an open subset $j:U\hookrightarrow Y$.
Let $g:X\to U$ be the restriction of $\overline{f}$ to $X$. Since $g_*\mathcal{O}=\mathcal{O}$, the projection formula shows that $g_*g^*\mathcal{F}\simeq \mathcal{F}$ if $\mathcal{F}$ is coherent on $U$. As a consequence, if $\mathcal{F}$ is coherent on $U$, $j_*\mathcal {F}=f_*g^*\mathcal{F}$, so that $j_*$ preserves coherence. We have seen above that preserving coherence is local on the base. Localizing at a point in $X\setminus U$, we get that a morphism associated to the inclusion of an integral noetherian local ring of dimension $1$ in its field of fractions preserves coherence. This cannot hold, even for the structural sheaf $\mathcal{O}$.