One necessary condition (that suffices in your SU(2) case, and all others I could think of) is that the center of $G$ needs to act trivially in $V$ for $\mathrm{Hom}(W, W \otimes V)$ to ever be non-trivial. The character of the center justs multiplies in a tensor product, and so we can't have a map from $W$ to $V \otimes W$ if $V$ has a non-trivial central character. (This also follows from Ben's observation above.)
I don't know if this is also sufficient. This boils down to looking at groups with trivial center.