Born in Edmonton, Alberta, Canada, Morrison gained degrees in nutrition and pharmacology from the University of Alberta and Cornell University. He was an academic at the University of Guelph in Ontario and was chair of the Scientific and Technical Advisory Committee to the Special Program for Research and Training in Tropical Diseases in the World Health Organization. In 1984, Morrison was the recipient of the David M. Kennedy International Service Award from the Kennedy International Center at Brigham Young University. Morrison was also a deputy minister with Health Canada and oversaw issues of environmental and food safety.

During his time as a student at the University of Alberta, Morrison was baptized into The Church of Jesus Christ of Latter-day Saints after being introduced to Mormonism by a friend. At the time, Morrison's branch president was N. Eldon Tanner and his Sunday School instructor was Hugh B. Brown.

In 1987, Morrison became a member of the First Quorum of the Seventy of the LDS Church. Prior to his call as a general authority, Morrison had served as a church branch president, bishop, and Regional Representative of the Twelve Apostles. In 1989, Morrison was transferred to the newly-created Second Quorum of the Seventy, but was returned to the First Quorum in 1991. He served as a general authority until 2000, when he was given general authority emeritus status.

Morrison and his wife are the parents of eight children.

Morrison is the author of seven books on topics related to Mormonism and has become one of Mormonism's foremost voices on the plight of the mentally ill.