Sullivan was born in Huntington, West Virginia and attended public schools in that state. He graduated from Marshall University in 1968, and became a commissioned officer in the United States Army that same year. He was a decorated combat leader in Vietnam, and then returned to graduate school in Chapel Hill, starting in late 1970. He received an MBA in Finance in 1972, and started his career as a Division Controller with Burlington Industries.
Sullivan purchased a company of his own in 1979 and operated Dorsett Printing Corporation, a manufacturer of packaging products, for the next 30 years. Sullivan became active in community affairs in Lexington, North Carolina, with special interests in public education, economic development, and healthcare. In May, 2021 Sullivan was recognized for almost 50 years of community service and became only the eleventh person to receive an honorary degree from Davidson-Davie Community College in its 58 year history. Sullivan also served as CEO of the Lexington Area Chamber of Commerce for eight years after the sale of his company in 2009.
Burr and Rebecca Sullivan have been married for more than 51 years and have three sons and three grandsons. Rebecca is a retired public school teacher, and is very involved in community affairs. They both enjoy traveling, reading, spending time with family, and time devoted to community activities.