Thin-film technologies such as Indium Gallium Zinc Oxide (IGZO) enable Flexible Electronics (FE) for emerging applications in wearable sensing, personal health monitoring, and large-area systems. Analog-to-digital converters (ADCs) serve as critical sensor interfaces in these systems, yet their vulnerability to manufacturing defects remains poorly understood despite IGZO's inherently high defect densities and process variations compared to mature CMOS technologies. We present a hierarchical fault injection framework to characterize defect sensitivity in Binary Search ADCs implemented in FE technologies. Our methodology combines transistor-level defect characterization with system-level fault propagation analysis, enabling efficient exploration of both single and multiple fault scenarios across the conversion hierarchy. The framework identifies critical fault-sensitive circuit components and enables selective redundancy strategies targeting only the most sensitive components. The resulting defect-tolerant designs improve fault coverage from 60% to 92% under single-fault injections and from 34% to 77.6% under multi-fault injection, while incurring only 4.2% area overhead and 6% power increase.