Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then $\mathcal{F} = i_*\mathcal{G}$ for some coherent $\mathcal{O}_Z$-module $\mathcal{G}$ (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-coherent-support-closed}) and we have $$ \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) = \chi(Z, \mathcal{G} \otimes i^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes i^*\mathcal{L}_r^{\otimes n_r}) $$ by the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}) and Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-relative-affine-cohomology}. Since $|Z| = \text{Supp}(\mathcal{F})$ we see that it suffices to show $$ P_\mathcal{F}(n_1, \ldots, n_r) : (n_1, \ldots, n_r) \longmapsto \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree at most $\dim(X)$. Let us say property $\mathcal{P}$ holds for the coherent $\mathcal{O}_X$-module $\mathcal{F}$ if the above is true. \medskip\noindent We will prove this statement by devissage, more precisely we will check conditions (1), (2), and (3) of Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant} are satisfied. \medskip\noindent Verification of condition (1). Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. By Lemma \ref{lemma-euler-characteristic-additive} we have $$ P_{\mathcal{F}_2}(n_1, \ldots, n_r) = P_{\mathcal{F}_1}(n_1, \ldots, n_r) + P_{\mathcal{F}_3}(n_1, \ldots, n_r) $$ Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$ have property $\mathcal{P}$, then so does the third. \medskip\noindent Condition (2) follows because $P_{\mathcal{F}^{\oplus m}}(n_1, \ldots, n_r) = mP_\mathcal{F}(n_1, \ldots, n_r)$. \medskip\noindent Proof of (3). Let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. We have to find a coherent module $\mathcal{G}$ on $X$ whose support is $Z$ such that $\mathcal{P}$ holds for $\mathcal{G}$. We will give two constructions: one using Chow's lemma and one using a finite cover by a scheme. \medskip\noindent Proof existence $\mathcal{G}$ using a finite cover by a scheme. Choose $\pi : Z' \to Z$ finite surjective where $Z'$ is a scheme, see Limits of Spaces, Proposition \ref{spaces-limits-proposition-there-is-a-scheme-finite-over}. Set $\mathcal{G} = i_*\pi_*\mathcal{O}_{Z'} = (i \circ \pi)_*\mathcal{O}_{Z'}$. Note that $Z'$ is proper over $k$ and that the support of $\mathcal{G}$ is $Y$ (details omitted). We have $$ R(\pi \circ i)_*(\mathcal{O}_{Z'}) = \mathcal{G} \quad\text{and}\quad R(\pi \circ i)_*(\pi^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) ) = \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r} $$ The first equality holds because $i \circ \pi$ is affine (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}) and the second equality follows from the first and the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}). Using Leray (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}) we obtain $$ P_\mathcal{G}(n_1, \ldots, n_r) = \chi(Z', \pi^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r})) $$ By the case of schemes (Varieties, Lemma \ref{varieties-lemma-numerical-polynomial-from-euler}) this is a numerical polynomial in $n_1, \ldots, n_r$ of degree at most $\dim(Z')$. We conclude because $\dim(Z') \leq \dim(Z) \leq \dim(X)$. The first inequality follows from Decent Spaces, Lemma \ref{decent-spaces-lemma-dimension-quasi-finite}. \medskip\noindent Proof existence $\mathcal{G}$ using Chow's lemma. We apply Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow} to the morphism $Z \to \Spec(k)$. Thus we get a surjective proper morphism $f : Y \to Z$ over $\Spec(k)$ where $Y$ is a closed subscheme of $\mathbf{P}^m_k$ for some $m$. After replacing $Y$ by a closed subscheme we may assume that $Y$ is integral and $f : Y \to Z$ is an alteration, see Lemma \ref{lemma-alteration-contained-in}. Denote $\mathcal{O}_Y(n)$ the pullback of $\mathcal{O}_{\mathbf{P}^m_k}(n)$. Pick $n > 0$ such that $R^pf_*\mathcal{O}_Y(n) = 0$ for $p > 0$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-kill-by-twisting}. We claim that $\mathcal{G} = i_*f_*\mathcal{O}_Y(n)$ satisfies $\mathcal{P}$. Namely, by the case of schemes (Varieties, Lemma \ref{varieties-lemma-numerical-polynomial-from-euler}) we know that $$ (n_1, \ldots, n_r) \longmapsto \chi(Y, \mathcal{O}_Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r})) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree at most $\dim(Y)$. On the other hand, by the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}) \begin{align*} i_*Rf_*\left( \mathcal{O}_Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r})\right) & = i_*Rf_*\mathcal{O}_Y(n) \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r} \\ & = \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r} \end{align*} the last equality by our choice of $n$. By Leray (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}) we get $$ \chi(Y, \mathcal{O}_Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r})) = P_\mathcal{G}(n_1, \ldots, n_r) $$ and we conclude because $\dim(Y) \leq \dim(Z) \leq \dim(X)$. The first inequality holds by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-alteration-dimension-general} and the fact that $Y \to Z$ is an alteration (and hence the induced extension of residue fields in generic points is finite).