Dillard's BBQ was noted in this Indy article (thank you Indy for maintaining your URLs rather than switching them around all the time) as a "Durham staple since 1953." It certainly was among the echoes of a previous age of BBQ and Durham restaurants - a la Bullock's.
The restaurant closed in March of 2011. In December of 2013, the building was demolished. Tax records indicate, as of December 2013, that the parcel of owned by MM Fowler of Family Fare Fame.