Suvrit has answered it completely, but let me suggest how you might go about finding counterexamples.

It's often useful to work with not the Rényi entropies but their exponentials,
$$
D_\alpha(X) = \exp(H_\alpha(X)) = \Bigl( \sum_{i=1}^n p_i^\alpha \Bigr)^{1/(1-\alpha)}
$$
(where, as in your question, $X$ is a random variable with distribution $p_1, \ldots, p_n$). One advantage of working with $D$ rather than $H$ is that there's a useful limit as $\alpha \to \infty$, namely
$$
D_\infty(X) = 1/\max_i p_i.
$$
Since this is such a simple formula, $\alpha = \infty$ is a good case to try when testing conjectures.

In terms of $D$, subadditivity becomes $D_\alpha(A, B) \leq D_\alpha(A) D_\alpha(B)$. It's easy to find counterexamples when $\alpha = \infty$: for instance,
$$
\begin{pmatrix}
1/2 &1/4 \\\
1/4 &0
\end{pmatrix}
$$
is a counterexample since
$$
\frac{1}{\max\{1/2, 1/4, 1/4, 0\}} = 2 > \frac{16}{9} = \frac{1}{\max\{3/4,1/4\}}\frac{1}{\max\{3/4,1/4\}}.
$$
It follows that this is a counterexample for all sufficiently large finite $\alpha$.
If you graph it, you see that it is in fact a counterexample for all $\alpha$ greater than about $1.6$. Tweaking it gives you counterexamples for all $\alpha > 1$.