As charities say, it takes money to raise money -- and a growing number of organizations are tapping donors for those costs through what's known as a "gift tax."

Big donors often want their entire gift to go to a specific purpose, be it a research program, a construction project or something else. To keep them satisfied, many organizations exempted those donations from being used, to any extent, for fundraising and other general administrative costs. These costs were covered by endowment profits or unrestricted gifts.

"In...