It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $i = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} (combined with Topology, Lemma \ref{topology-lemma-topology-quasi-separated-scheme}). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$. \medskip\noindent By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_X$-modules $(\mathcal{F}_i, \varphi_{ii'})$ over $I$ is directed. By Injectives, Lemma \ref{injectives-lemma-sheaves-modules-space} the category of $\mathcal{O}_X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_i, \varphi_{ii'})$ there exists a system $(\mathcal{I}_i, \varphi_{ii'})$ with each $\mathcal{I}_i$ an injective $\mathcal{O}_X$-module and a morphism of systems given by injective $\mathcal{O}_X$-module maps $\mathcal{F}_i \to \mathcal{I}_i$. Denote $\mathcal{Q}_i$ the cokernel so that we have short exact sequences $$ 0 \to \mathcal{F}_i \to \mathcal{I}_i \to \mathcal{Q}_i \to 0. $$ We claim that the sequence $$ 0 \to \colim_i \mathcal{F}_i \to \colim_i \mathcal{I}_i \to \colim_i \mathcal{Q}_i \to 0. $$ is also a short exact sequence of $\mathcal{O}_X$-modules. We may check this on stalks. By Sheaves, Sections \ref{sheaves-section-limits-presheaves} and \ref{sheaves-section-limits-sheaves} taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}) we deduce the result. We claim that $H^q(U, \colim_i \mathcal{I}_i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram $$ \xymatrix{ \colim_i H^{q_0}(U, \mathcal{I}_i) \ar[d] \ar[r] & \colim_i H^{q_0}(U, \mathcal{Q}_i) \ar[d] \ar[r] & \colim_i H^{q_0 + 1}(U, \mathcal{F}_i) \ar[d] \ar[r] & 0 \ar[d] \\ H^{q_0}(U, \colim_i \mathcal{I}_i) \ar[r] & H^{q_0}(U, \colim_i \mathcal{Q}_i) \ar[r] & H^{q_0 + 1}(U, \colim_i \mathcal{F}_i) \ar[r] & 0 } $$ The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_i$ are injective. The top row is exact by an application of Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}. Hence by the snake lemma we deduce the result for $q = q_0 + 1$. \medskip\noindent It remains to show that the claim is true. We will use Lemma \ref{lemma-cech-vanish-basis}. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup_{j = 1, \ldots, m} U_j$ with each of $U$, $U_j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i) = \colim_i \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_i) $$ because all the multiple intersections $U_{j_0 \ldots j_p}$ are quasi-compact. By Lemma \ref{lemma-injective-trivial-cech} each of the complexes in the colimit of {\v C}ech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact} we see that also the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^p(\mathcal{U},  \colim_i \mathcal{I}_i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma \ref{lemma-cech-vanish-basis} are satisfied and the claim follows.