My example is similar to Greg's, and shows that $H^i(X,M), H^i(X,L)$ can vanish while $H^i(X, M \otimes L)$ doesn't have to for $X$ the affine line over a finite field. Let $\psi \colon \mathbb{F}_q \to \overline{\mathbb{Q}}_\ell^\times$ be a non trivial character. The Artin-Schreier cover of the affine line $\mathbb{A}^1$ over $\mathbb{F}_q$ is given by
$$ p \colon \mathbb{A}^1 \to \mathbb{A}^1, \quad x \mapsto x^q - x. $$
Hence there is a surjection $\pi_1^\text{et}(\mathbb{A}^1) \to \mathrm{Aut}(p) \simeq \mathbb{F}_q$. Let $\mathcal{L}(\psi)$ be the lisse sheaf associated to the representation
$$ \pi_1^\text{et}(\mathbb{A}^1) \to \mathbb{F}_q \xrightarrow{\psi} \overline{\mathbb{Q}}_\ell^\times $$
If $x \in \mathbb{F}_q$ write $\psi_x$ for the character $y \mapsto \psi(xy)$. Then $p_\ast \overline{\mathbb{Q}}_\ell \simeq \oplus_{x \in \mathbb{F}_q} \mathcal{L}(\psi_x)$. From the Leray spectral sequence, we get that $H^i(\mathbb{A}^1, \mathcal{L}(\psi_x)) = 0$ unless $i = 0$ and $x = 0$. Now given $x \in \mathbb{F}_q^\times$ we have $\mathcal{L}(\psi_x) \otimes \mathcal{L}(\psi_{-x}) \simeq \overline{\mathbb{Q}}_\ell$.
So summing up, for any $x \in \mathbb{F}_q^\times$ we have $R\Gamma(\mathbb{A}^1, \mathcal{L}(\psi_x)) = 0$ but $R\Gamma(\mathbb{A}^1, \mathcal{L}(\psi_{x}) \otimes \mathcal{L}(\psi_{-x})) = \overline{\mathbb{Q}}_\ell$. A similar argument shows also that $R\Gamma_c(\mathbb{A}^1, \mathcal{L}(\psi_x)) = 0$ but $R\Gamma_c(\mathbb{A}^1, \mathcal{L}(\psi_{x}) \otimes \mathcal{L}(\psi_{-x})) = \overline{\mathbb{Q}}_\ell[2]$.