What important psychological results would not have been discovered without experiments on non-human animals?
I think this necessarily involves some speculation. It's very hard to tell whether certain important findings would have been discovered without studies in non-human animals. We should also keep the hindsight bias in mind, that is, the tendency to retroactively claim that we 'knew it all along'. 

Another aspect to consider is that only experiments enable us to investigate the causality of a relationship that we might have already observed in a non-experimental context. For example, we might have already observed that people with depression also show signs of helplessness, but this usually isn't enough to assume a causal link between these two.

Finally, I do think there are a few findings that would have been almost impossible without research in non-human animals. For example, any knowledge that stems from the recording of individual neurons in the brain. It's currently infeasible to record data from individual neurons in humans (apart from a few patients that have medical devices implanted due to neurological disorders). So for example, our knowledge of dopamine neurons signaling reward prediction errors depended critically on research in non-human animals (Schultz et al., 1997, Science).