In answering your question, I am making three assumptions: (1) that all the directors have previously passed away and you are not merely assuming that they will die before the end of the lease (if that were the case, the living directors could simply appoint new directors even if there is less than a quorum of the board still alive); (2) that the nonprofit has no members who can elect new directors, and (3) that the owner “nonprofit” is a 501(c)(3) charity, which may affect the ultimate disposition of the property.  (See Ready Reference Page: “What Do We Mean When We Say Nonprofit”?)

This is an unusual situation that isn’t even possible in the for-profit world.  With a business entity, there are owners whose economic interest passes by will or intestacy when they die, so somebody will always have the power to select people to govern the organization.  In the nonprofit world, if there are no members or other directors who can act, the rights of directors cannot be inherited, and you can end up with an entity that has nobody responsible for it.   

The first thing I would do is find the governing documents of the organization to see if there is some person or entity still extant that has some role in the governance of the organization and would have the power to appoint new directors.  If that fails, I would consult with the state attorney general about getting court approval (if the AG says it is necessary) for the appointment of a new board to take responsibility for the organization, even if the new directors are only empowered to dissolve it.  (Who has legal standing to deal with the situation may be unclear, but the AG normally has jurisdiction to deal on behalf of the public with any charity.  The AG may be willing to deal with representatives of the tenant as the people who have the most interest in the charity and its assets.)

The organization obviously has some assets, particularly the leased property. The new board could continue the lease, but if the organization does not continue to function and dissolves, its net remaining assets must be distributed for charitable purposes.  Its governing documents may specify where the assets are to go upon dissolution, but if not, the new directors could probably decide.  They might be able to give the property to the tenant if the tenant is a charity, but probably couldn’t if the tenant is not a charity, such as a social club or a (c)(4) social welfare organization.  They might be able to sell it to the tenant at fair market value and use the proceeds for other charitable purposes, however.

The key to all of this is getting a new board appointed, and going through the attorney general seems the most likely route if there is nobody else around who has any legal authority in the situation.