Abner is engaged to actress Hallie Foote and appeared in Alone alongside Foote. His father-in-law, Horton Foote, wrote the stage play "The Trip to Bountiful" in which Abner portrayed the role of Ludie Watts in 2003 and between 2005 and 2006. In 2006 he received a Drama Desk Award nomination for best featured actor for his performance in this play.
Other appearances include the television drama Alone (1997, with Ed Begley, Jr., Joanna Miles, and Sarah Freeman), The Office episode Halloween (2005), and the television movie Love's Unfolding Dream (2007, with Robert Pine, Richard Herd, and J.C. Brandy).