I am familiar with a few commercial-type nonprofits that utilize non-compete provisions for employees, and you can read a lot of cases about non-compete provisions for physicians employed by nonprofits, especially hospitals that subsidize a physician in establishing a practice in an underserved area.  I am not familiar with non-competes being required for board members who are not direct service providers, however, and I would be skeptical about the need or desirability of such an agreement.

Whether it would be enforceable depends on whether there is actually consideration for the agreement, and, if so, whether it is reasonable as to the area covered, types of activities covered, duration, and any other limitations, such as being prevented from being on the board of a different organization that provides such services as well as being prevented from personally providing them.

If I were more interested in assuring that the services are available in the community than in assuring that only this organization or this leader would be in a position to provide them, I think I would decline the opportunity to sign the non-compete.

UPDATE: 6/2024

The recently approved Final Rule of the Federal Trade Commission prohibiting non-compete agreements in most employment situations applies to corporations only if they are "organized to carry on business for its own profit or that of its members."  But in the notice announcing the Rule, the Commission recites precedent applying FTC rules to nonprofit entities "if they are in fact profit-making enterprises."  Most of the exceptions to the exemption appear to be in medical situations such as hospitals or physicians' groups, but the exact contour of the rule will have to be decided by later decisions.

The Comments below were submitted prior to the FTC's final Rule.