Greer was the son of Dudley Jefferson Greer and his third wife, the former Sarah Holder. They moved to Birmingham in 1907 where Dudley Greer worked for Robert Stewart's House of Robert building firm. They returned to South Carolina in 1912. Sarah moved back to Birmingham with Dupree after Dudley's death in 1920 and married Stewart.
Greer began his career as a draftsman for architect H. G. Little and later worked for Bechtel-McCone-Parsons. He was hired as a chief draftsman and inspector by E. B. Van Keuren and J. W. Goodwin. He briefly had his own office before the Great Depression, then worked for the U.S. Department of the Interior from 1934 until joining Walter Holmquist and William Chambers in 1938.
- American Architects Directory (1962) American Institute of Architects