This CSL ("converse of Schur's Lemma") condition on a ring is actually a topic of interest in ring theory lately. This basically means that there is not yet any simple answer to the question. But there is some interesting progress toward partial answers. For instance, a recent paper by G. Marks and M. Schmidmeier shows that the converse of Schur's Lemma holds in the category of right R-modules of finite length if and only if all extensions of simple right R-modules are split. This holds, in particular, over any commutative ring.
Those authors also show that a semiprimary ring R (that is, R has a nilpotent Jacobson radical J, and R/J is semisimple) satisfies CSL on the right if and only if all extensions of simple right R-modules are split, if and only if R is a finite direct product of matrix rings over local rings. (Examples of semiprimary rings include one-sided artinian rings, such as finite dimensional algebras over fields.)
The same paper cites a number of other sources if you are interested in further exploring the topic. For instance, there are references for the following result, similar to the one above: A one-sided noetherian ring has CSL on the right if and only if it is a finite direct product of matrix rings over local perfect rings (which must be one-sided noetherian, hence one-sided artinian).