Nonstock corporations, as they are called in Virginia, do not pay income taxes because they are formed for non-personal profit purposes.
By filing proper state and federal documents, they may receive tax-deductible contributions and grant funds from other tax-exempt public and private agencies. While they may not payout profits to its members, they may pay them reasonable salaries. Also, upon dissolution, their assets do not simply revert to the contributing members. They are distributed to other tax-exempt organizations.
As with regular corporations, directors and members have limited liability for debts and liabilities of nonstock corporations, and these corporations exist in perpetuity.