Henry Luce III, the elder son of the founder and editor in chief of Time Inc., who held prominent positions at the company and went on to be head of the Henry Luce Foundation, died on Wednesday at his summer home on Fishers Island, N.Y. He was 80. The cause was not immediately determined, said Terry Lautz, vice president of the foundation. Henry Luce III, the son of Henry R. Luce, was the grandson of Henry Winters Luce, who, with his wife, Elizabeth, was an educational missionary in China. Henry III, known by the nickname Hank, had no middle name.