= Solution
Recall that a <flasque sheaf> has surjective restriction maps on every inclusion of open subsets. We first prove the <flasque-kernel section-lifting lemma>: in an exact sequence $0\to\mathcal A\to\mathcal B\to\mathcal C\to0$ of <sheaves>, with $\mathcal A$ a <flasque sheaf>, every section of $\mathcal C$ on any open subset $V$ lifts to a section of $\mathcal B$ on $V$.
Fix such a section $c$. Order its lifts on open subsets $W\subseteq V$ by extension. A chain has an upper bound by the <sheaf gluing axiom>, so <Zorn's lemma> gives a maximal lift $b_W$. If $W\ne V$, choose a point outside $W$ and a neighborhood $T\subseteq V$ on which $c$ has a local lift $b_T$. On $W\cap T$, the difference $b_T-b_W$ is a section of $\mathcal A$. Extend it to $T$ by <flasqueness> and subtract it from $b_T$. The corrected lift now agrees with $b_W$ and glues on $W\cup T$, contradicting maximality. Thus $W=V$, proving the lemma.
It follows also that if both $\mathcal A$ and $\mathcal B$ are <flasque sheaves>, then $\mathcal C$ is a <flasque sheaf>. Given a section of $\mathcal C$ on $U\subseteq V$, first lift it to $\mathcal B$ on $U$, extend that lift to $V$, and project to $\mathcal C$.
Next, an <injective sheaf of modules> $\mathcal I$ is a <flasque sheaf>. For $U\subseteq V$, let $j_U$ and $j_V$ denote the open inclusions into $X$. <Extension by zero> gives a monomorphism $j_{U!}\mathcal O_U\to j_{V!}\mathcal O_V$: on every <stalk> it is either $0\to\mathcal O_{X,x}$, the identity, or $0\to0$. By the adjunction for <extension by zero>,
$$
\operatorname{Hom}_{\mathcal O_X}(j_{U!}\mathcal O_U,\mathcal I)=\Gamma(U,\mathcal I).
$$
Injectivity therefore makes $\Gamma(V,\mathcal I)\to\Gamma(U,\mathcal I)$ surjective. This proves <injective module sheaves are flasque>.
Take an <injective resolution of sheaves> of $\mathcal F$. Write $\mathcal C^0=\mathcal F$ and, successively,
$$
0\longrightarrow\mathcal C^r\longrightarrow\mathcal I^r\longrightarrow\mathcal C^{r+1}\longrightarrow0.
$$
The initial $\mathcal C^0$ is a <flasque sheaf>; each $\mathcal I^r$ is a <flasque sheaf>; and the quotient argument proves inductively that all $\mathcal C^r$ are <flasque sheaves>. The section-lifting lemma makes each displayed sequence exact after <global sections>. Consequently the complex of <global sections> of the <injective resolution of sheaves> has zero positive cohomology. Since that complex defines <sheaf cohomology>,
$$
\boxed{H^p(X,\mathcal F)=0\quad(p>0).}
$$
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