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My favorite (counter)example is this: let $A$, $B$, $C$ be the categories of left modules over some rings $R$, $S$, $T$ (respectively), and let $F$ and $G$ be the functors of tensor product with some bimodules, that is $F(M)=K\otimes_RM$ and $G(N)=L\otimes_SN$, where $K$ is an $S$-$R$-bimodule and $L$ is a $T$-$S$-bimodule. Then the left derived functor $\mathbb D (GF)$ is the functor of derived tensor product with the underived tensor product $L\otimes_SK$, while the composition of derived functors $\mathbb D (G)\mathbb D(F)$ is the functor of derived tensor product with the derived tensor product $L\otimes_S^{\mathbb D}K$, that is $\mathbb D(GF)(M)=(L\otimes_SK)\otimes_R^{\mathbb D}M$ and $\mathbb D(G)\mathbb D(F)(M)=L\otimes_S^{\mathbb D}K\otimes_R^{\mathbb D}M$.