What rights does the founder of a non-profit have if he leaves the organization?

When at odds with the board of directors and feeling stressed and harassed, after running a very successful operation, the founder and executive director resigned. Does the founder have any right to be recognized as the founder of the organization while the business continues?

A founder of an organization does not, as a matter of law, have a specific right to continue to be recognized as such unless there are specific terms in the certificate of incorporation of the bylaws addressing that (which there usually aren't).

Like for-profit corporations, the focus is the business--or in your case, the people or causes being served. Unless there is a culture dedicated to preserving the legacy, there often is nothing in the law that says that the founder must continue to be recognized as such.

As a matter of marketing, however, it probably would be good form to recognize the substantial dedication that the founder brought to the organization in getting it started and running it so successfully over the years. It could make the difference between the friends of the founder continuing to support the organization or finding another worthy cause to contribute to.