RALEIGH, N.C.—There will be no live opossum at an Appalachian community’s Possum Drop this New Year’s Eve, after concerns by animal-rights groups spurred an agreement between the festival organizer and a state administrative law judge.

The community of Brasstown, population 240, has been lowering a plexiglass box containing a live opossum from a pole at Clay’s Corner Convenience Store at midnight on Dec. 31 for the better part of 20 years. Organizers call the Brasstown Possum Drop a hillbilly homage to the ball drop in Times...