Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules: $$ \cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow I_4 \overset{f_3}\longrightarrow I_3 \overset{f_2}\longrightarrow I_2\overset{f_1}\longrightarrow I_1 \overset{f_0}\longrightarrow 0\tag{*} $$ Now, given an element $i\in \Gamma(X, I_n)$ with $f_{n-1}(i) = 0$ (or, rather, $f_{n-1, X}$), is there any hope of finding a $j \in \Gamma(X, I_{n+1})$ such that $f_n(j) = i$? If assuming $X$ to be Noetherian is necessary, then that is fine.

Partial answer: if $X$ has underlying topological space of finite dimension $k$, then what you want is true, because $H^i(X, F)=0$ for any $O_X$-module $F$ and $i>k$. Let $F=\ker(f_{k+n})$, so that we have an injective resolution $$ 0\to F\to I_{n+k+1} \to I_{n+k} \to \ldots \to I_{n+1}\to I_n \to I_{n-1} \to\ldots $$ So if your $j$ didn't exist, $i$ would give a nonzero class in $H^{k+1}(X, F)=0$.