Interposer-based 2.5D integrated circuits (ICs) are seen today as a precursor to 3D ICs based on through-silicon vias (TSVs). All the dies and the interposer in a 2.5D IC must be adequately tested for product qualification. We present an efficient built-in self-test (BIST) architecture for targeting defects in dies and in the interposer interconnects. The proposed BIST architecture can also be used for fault diagnosis during interconnect testing. We present simulation results to validate the BIST architecture and demonstrate fault detection, synthesis results to evaluate the area overhead of the proposed BIST architecture, and fault coverage results to highlight the effectiveness of the proposed technique.