What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that the right derived functor $$ RF:\text D(\mathcal C)\to\text D(\mathcal C') $$ does not exist?
My reference for the notions involved in this post is the book Categories and Sheaves by Kashiwara and Schapira.
Here is a reminder of the definition of a right derived functor $RF$ in the above setting: Let $\text K(F):\text K(\mathcal C)\to\text K(\mathcal C')$ be the triangulated functor induced by $F$ between the homotopy categories, and let $Q:\text K(\mathcal C)\to\text D(\mathcal C)$ and $Q':\text K(\mathcal C')\to\text D(\mathcal C')$ be the localization functors. A left Kan extension $RF$ of $Q'\circ\text K(F)$ along $Q$ (assuming that such exists) is a right derived functor for $F$ if, for any functor $G:\text D(\mathcal C')\to\mathcal A$, the functor $G\circ RF$ is, in a natural way, a left Kan extension of $G\circ Q'\circ\text K(F)$ along $Q$.
Here is an equivalent definition: Assume that for all $X$ in $\text K(\mathcal C)$ the colimit $$ \operatorname*{colim}_{X\to Y}\ (Q'\circ\text K(F))(Y), $$ where $X\to Y$ runs over all the quasi-isomorphisms out of $X$ in $\text K(\mathcal C)$, exists in $\text D(\mathcal C')$, and denote this colimit by $RF(X)$. Then $RF$ is a right derived functor for $F$ if, for any functor $G:\text D(\mathcal C')\to\mathcal A$, the colimit $$ \operatorname*{colim}_{X\to Y}\ (G\circ Q'\circ\text K(F))(Y) $$ exists in $\mathcal A$ and coincides with $G(RF(X))$ for all $X$ in $\text K(\mathcal C)$.