Well, as already explained in the comments by Vivek Shende and Allen Knutson, the answer is no: there are non-proper morphisms with push-forward that preserves coherence. I've thought a bit on this the last couple of years and on the "positive" side we have:
Theorem 1. If $f\colon X\to Y$ is a universally closed morphism of finite type between noetherian schemes, then $f_*$ preserves coherence.
This is quite easy to show with Raynaud–Gruson's refined Chow Lemma (also when $X$ and $Y$ are algebraic spaces) and standard dévissage arguments. With a completely different proof I have also extended this to algebraic stacks.
$\mathbb{P}^2\setminus 0\to *$ does not preserve coherence as there exist non-proper curves. If $f_*$ preserves coherence after arbitrary noetherian base change, then $f$ is universally closed: the valuative criterion is easily seen to hold (as indicated by Piotr Achinger) since the fraction field of a DVR is not coherent. Actually, using Riemann–Zariski tricks, it seems that we have:
Theorem 2. Let $f\colon X\to Y$ be a morphism of finite type between noetherian schemes. Then $f_*$ preserves coherence if and only if $f$ is universally closed.
Proof. (sketch) One direction is Theorem 1. For the other direction, assume that $f$ is not universally closed. Then there exists a DVR $D$ and a morphism $g\colon \operatorname{Spec} D\to Y$ and a closed immersion $\operatorname{Spec} K(D) \to X\times_Y \operatorname{Spec} D$. We may also assume that $K(D)$ is the residue field of the image of the generic point in $Y$. Replacing $Y$ with the closure of the image of $g$ we may assume that $Y$ is integral with $K(D)=K(Y)$.
Note that a dominant affine open immersion does not preserve coherence unless it is an isomorphism (if it preserves coherence it is finite hence a closed and open immersion, hence an isomorphism).
Now, by the theory of Riemann–Zariski spaces, the DVR $D$ is a local ring of the inverse limit of all blow-ups on $Y$ (when $Y$ is of finite type over a field, and $D$ is "algebraic" then a finite number of blow-ups is sufficient, cf. Kollar–Mori Lem. 2.45). By approximation, this means that we can find
(i) a blow-up $Y'\to Y$;
(ii) an affine open immersion $U'\to Y'$;
(iii) an affine open immersion $V'\to U'$; and
(iv) a closed immersion $V'\to X\times_Y U'$.
The morphism $\operatorname{Spec} D\to Y$ factors through $U'$ and $\operatorname{Spec} K(D)\to \operatorname{Spec} D$ is the pull-back of $V'\to U'$ etc.
Now consider the closure $\overline{V'}$ in $X\times_Y Y'$. Then $V'=\overline{V'}\times_{Y'} U'$. This means that the push-forward of the structure sheaf of $\overline{V'}$ to $Y'$ is not coherent (since the restriction to $U'$ is not coherent).
Finally, we pick a relatively ample line bundle $\mathcal{O}_{Y'}(1)$ for $Y'\to Y$. If we let $F=\mathcal{O}_{\overline{V'}}(n)\in \mathrm{Coh}(X\times_Y Y')$ and pick a large $n$, then the push-forward to $Y'$ is not coherent and generated by global sections relative to $Y$. This means that the push-forward to $Y$ is not coherent. QED.
Remark 1. In basic non-separated examples $R^if_*$ do not preserve coherence and perhaps coherence of higher direct push-forwards does imply separatedness as Tom Graber says. For schemes and algebraic spaces, Raynaud–Gruson's Chow lemma essentially reduces this question to morphisms of the form $f\colon X\to \mathbb{P}^n_Y \to Y$ where $f\colon X\to \mathbb{P}^n_Y$ is étale. Twisting, this perhaps reduces the question to étale morphisms. This approach could perhaps also give a simpler proof of Theorem 2.
Remark 2. The situation for algebraic stacks is more subtle. Theorem 1 holds as mentioned above. Theorem 2 can be extended to stacks as well: we can at least let $Y$ be Deligne–Mumford (or have quasi-finite and separated diagonal) and $X$ be arbitrary. One immediately reduces to $Y$ a scheme by taking a finite flat covering. The same proof then works with the only difference that $V'\to U'$ becomes some morphism followed by an affine open immersion (hence the push-forward of the structure sheaf is not coherent). This is enough for the proof to work. Remark 1 is more subtle though. If Remark 1 holds for schemes/algebraic spaces and $X$ has a good moduli space then $X$ has coherent cohomology if and only if $X_\mathrm{gms}$ is proper, e.g., BGL_n has coherent cohomology but is not separated. This is all well-known and one may argue that such stacks are "almost proper" (cf. paper by Halpern-Leistner–Preygel, arXiv:1402.3204).