P. Deligne has constructed non-residually finite central extensions of some arithmetic groups, so for these the inner automorphism group is residually finite, see: P. Deligne. Extensions centrales non r ́esiduellement finies de groupes arithm ́etiques. CR Acad. Sci. Paris, s ́erie A-B, 287, 203–208, 1978.
See also the central extensions constructed by D. Toledo in "Projective varieties with non-residually finite fundamental group", PUBLICATIONS MATHÉMATIQUES DE L'IHÉS, Volume 77, Number 1, 103-119: these are fundamental groups of compact manifolds, hence clearly finitely presented.