I tried a lot of examples and the inverse nakayama functor $\nu^{-1}$ seems to be zero on non injective represetantions for all quivers I tried. Of course this is false if the quivers has relations (as I learned here ). But in the case of a quiver with no relations it seems to be always zero on non injectives. I even proved it for any quiver of the form $\mathbb{A}_n$ using a rudimentary method of studying the possible morphisms between an indescomposable injective $I(k)$ and any non injective indescomposable representation. And I think that the same is true for any quiver $\mathbb{D}_n$ but the proof seems way more complicated.
Does this hold in general for quivers with no relations (and possibly no cycles)?
In quivers with no relations and cycles, the inverse Nakayama functor $\nu ^{-1}$ maps indecomposable noninjective modules to zero. And it maps indecomposable injective modules to the corresponding indecomposable projective modules. For example: quiver $Q: 1 \rightarrow 2$, then $\nu ^{-1}(S(2))=0$ and $\nu^{-1}(S(1))=P(1)$, so $\nu^{-1}(S(1) \oplus S(2))=P(1)$