This functional is sequentially weakly lower semicontinuous under fairly mild assumptions on $f$.
We need that $f$ is nonnegative, continuous and bounded from above.
Let $u_n \rightharpoonup u$ in $H^1(\Omega)$ and $v_n \rightharpoonup v$ in $H_0^1(\Omega)$.
RellichKondrachov implies that $u_n \to u$ in $L^2(\Omega)$ (we might need a mild assumption on $\Omega$ here).
Hence, $f(u_n) \to f(u)$ (along a subsequence) a.e.
For $\varepsilon > 0$ we can use Egorov's theorem to get a subset $E \subset \Omega$
of measure smaller than $\varepsilon$ such that
$f(u_n) \to f(u)$ uniformly on $\Omega \setminus E$.
On $\Omega \setminus E$ we can use
\begin{equation*}
\int_{\Omega \setminus E} f(u_n) \, \nabla v_n^2 \, \mathrm{d}x
=
\int_{\Omega \setminus E} f(u) \, \nabla v_n^2 \, \mathrm{d}x.
+
\int_{\Omega \setminus E} (f(u_n)  f(u)) \, \nabla v_n^2 \, \mathrm{d}x
\end{equation*}
For the first addend, we can use weak lower semicontinuity,
whereas the second addend goes to zero.
On $E$ we use the simple estimate
\begin{equation*}
\int_{E} f(u_n) \, \nabla v_n^2 \, \mathrm{d}x \ge 0.
\end{equation*}
Alltogether, we get
\begin{equation*}
\liminf_{n \to \infty} J(u_n, v_n)
\ge
\int_{\Omega \setminus E} f(u) \, \nabla v^2 \, \mathrm{d}x
=
J(u,v)

\int_{E} f(u) \, \nabla v^2 \, \mathrm{d}x
.
\end{equation*}
Since $f(u) \, \nabla v^2 \in L^1(\Omega)$, we have
\begin{equation*}
\int_{E} f(u) \, \nabla v^2 \, \mathrm{d}x
\to
0
\end{equation*}
for $\varepsilon \searrow 0$.
This shows
\begin{equation*}
\liminf_{n \to \infty} J(u_n, v_n)
\ge
J(u,v)
.
\end{equation*}