Depends how the profit was earned. If it's through activities related to their non-profit status then it's still tax exempt, and they have to reinvest it into the company rather than distribute it to owners/shareholders/etc as with a for profit company. If the profit came from activities unrelated to their non profit status then that income may be subject to taxes.
Discussion
Revenue/profit unrelated to their non profit status also shouldn't be a huge portion of their finances. I think if it is that could potentially jeopardize their non profit status.