What is the causal relationship between extreme jealousy and poor mental health? Jealousy can be a powerful and destructive force. It can take over reason.

Can severe to extreme jealousy precipitate or contribute to poor mental health, at the extreme, psychosis.

Or, is it poor mental health that can manifest pathological jealousy?

or both.
I think it can go both ways. According to Tesser's Self Evaluation Maintenance Theory, any two people in a relationship make themselves feel better by comparing themselves to each other. The key idea is that if you are closer to a person then you will tend to feel more jealous. For example, suppose your best friend got a really fancy car. You will naturally feel jealous. However, what if someone told you that Bill Gates got a really fancy car? You wouldn't feel jealous.This is similar to mental construal theory mentioned in this post.

If you tend to be jealous of someone close to you all the time I would think this would lead to depression. Depression is marked by rumination which is what happens if you are constantly jealous. Then again, if you are depressed because of some acute stress (e.g. death in a family) then you could feel jealous of other people without this particular stress. 

One of the best ways of overcoming jealousy is to accept that you are jealous and empathize/congratulate the person you are jealous of. Exercises like mindfulness meditation would help promote empathetic behaviors.