An ordained Methodist minister, Robert Tappan Osborn was born in 1926 in Seattle, Washington. He received his A.B. from U.C.L.A. in 1946 and his B.D. from Garrett Biblical Institute in 1950, and spent a year at the University of Zurich, Switzerland. In 1954 he was awarded the Ph.D. from Drew University and joined the faculty of the Duke University Department of Religion that same year, where he taught until his retirement in 1997. He served a term as chair of that department. His research focused on the field of contemporary theology and Christian thought and his publications include the books (1967) and (1991).

Osborn became active in the civil rights movement during the 1960s, advocating for Christian involvement in issues of racial justice. In January 1964, he and four other Duke faculty members (Professor William Wynn of the Psychology Department, Professor Fred Herzog and Professor Harmon Smith, both of Duke Divinity School, and Professor Peter Klopfer of the Zoology Department) joined a group including students and faculty from UNC and NCCU (white and African American) for a restaurant sit-in at Watts Grill in Chapel Hill. The professors were arrested on counts of trespassing; and the ensuing, drawn-out trials in Hillsborough ultimately led to a U.S. Supreme Court decision in the case of . In the final days of his term as governor, Terry Sanford (later to become president of Duke University) pardoned the professors.