Given abelian categories $\mathcal A, \mathcal B$ and a left exact functor $F: \mathcal A \to \mathcal B$, we can define the notion of a right derived functor (or universal $\delta$-functor) by its universal property. Roughly speaking, we have functors $R^iF$ for any $i$ and short exact sequences give rise to long exact sequences in a functorial way. Furthermore the $R^iF$ are universal with this property, hence they are unique if they exist.
The following is well known:
If $\mathcal A$ has enough injectives, one can construct right derived functors via injective resolutions, hence they always exist and as an immediate consequence we have $R^iF(I)=0 ~\forall i > 0$ for any injective object $I$.
If the category does not have enough injectives (there might be some - but not enough - non-zero injective objects though), we still can ask the following question:
If the right derived functors of $F$ exist, does $R^iF(I)=0 ~ \forall i >0$ for any injective object $I$ still hold?
Morally the question is the following: In the "usual case" with enough injectives, is the construction of the right derived functors crucial to the acyclity of injective objects or is it a formal consequence of the universal property, regardless of the existence?
Note. The question was already posed here: Why do universal $\delta$-functors annihilate injectives?, but unfortunately nobody came up with an answer.