What is the term for a psychological effect which does not have a neurobiological/genetic basis? When a psychological condition is the result of irregular brain structures or neurotransmitter imbalances we say the condition has a neurobiological basis; this is analogous to the classic Nature in the Nature vs Nurture discussion. 

What is the term for a condition which does not have this sort of (apparent) neurobiological basis? This would be a condition caused by Nurture effects such as psychologically traumatic events. 

Is there a distinction? Technically both would induce physical changes in the brain, but I often see "neurobiological basis" used as if there were a different sort of basis for a condition.
As I alluded in the comments, I think your question is underpinned by a more fundamental worry of the "best level of description" for psychological phenomena. If you a reductionist then you believe that all of psychology is best explained in terms of the activity of neurons (or systems of neurons if you are of the system perspective; note that some do not consider this reductionist. However, it does reduce to the physical basis).

On the other hand, Putnam and Fodor (and I suspect most proponents of strong AI) argue for multiple realizability: mental states and psychology can be implemented in various physical substrates, not just biological neurons. If you subscribe to this view, then it is natural to expect the best level of description of psychology to be independent of the specifics of a physical implementation (think algorithm-analysis in theoretical computer science). Since brains are physical things that implement minds, you would believe that a neurobiological basis does exist, but its details are irrelevant to your theory. Note that many theories in psychology can be viewed at the macro- or behaviorist level and as I suggested earlier: many theories in cognitive science lack a solid neuronal grounding.

If you have a physics background, then the weak analogy I would make is thermodynamics versus statistical mechanics. Thermodynamics is a macro-level theory that can exist without micro-level detail. However, eventually thermodynamics found a micro-level grounding in statistical mechanics. Does that mean we abandon thermodynamics and only study stat mech?

As for your nature vs. nurture distinction, if we accept mutliple realizability then we can use the above ideas to try to formalize the terminology. We can think of nature features as ones that are best described at the level of the specific implementation of the human brain, and nurture features as ones that are best described on a macro-level that would apply to any reasonable implementation. I don't know if this idea has been deeply studied or if it has any interesting bearing on the nature vs. nurture debate.