What degree would I need to start my own mental health nonprofit?

If you just want to run it, and hire someone else to provide all the care, then maybe business or something.

If you want to provide some of the care yourself, then probably psychiatry or psychology.

None. But, you really need to dedictate yourself to getting a job.

Let's separate a few different things:

1. ANYONE can file a 501C and start a non-profit.

2. To manage any type of business or service, get loans and donations, develop contracts with providers and vendors, etc., a background in business and accounting is really helpful. If you do no have these skills/knowledge, you will have to hire someone who does.  

3. To actually BE a mental health counselor your minimum requirement would be a masters degree in counseling.  You could get a PhD in clinical or behavior psychology and or get an MD specializing in psychiatry.  

4. Depending on the services you want to provide, your organization would need licensed counselors, therapist, and psychiatrists to treat patients.  As the founder of the organization, you do NOT need to have those skills, but you'd have to hire people who do.

5.  There are some classes and courses of study that would also be helpful to anyone who is interested in working with underserved populations.  Those would include poverty studies, sociology, and public health.  It isn't enough to have services, you have to understand how the people you are trying to help think about and react to traditional options. Do they not have access to services or are they unwilling to get help?

Masters in Public Administration.

You personally won't be delivering any mental health services.   As Executive Director you will be creating policy, managing the finances/budget, marketing, fundraising, managing HR and reporting to the Board of Directors. 

If you want to be a provider of mental health services, then leave the administrative work to someone else.     There are a huge variety of mental health services.   If you want to practice psychiatry (be able to diagnose and prescribe), you need to go to medical school.   Or you could earn an MSW and become an LMFT.    You said nothing about what exactly you want to do.