Practical systems often suffer from hardware impairments that already appear during signal generation. Despite the limiting effect of such input-noise impairments on signal processing systems, they are routinely ignored in the literature. In this paper, we propose an algorithm for data detection and signal estimation, referred to as Approximate Message Passing with Input noise (AMPI), which takes into account input-noise impairments. To demonstrate the efficacy of AMPI, we investigate two applications: Data detection in large multiple-input multiple output (MIMO) wireless systems and sparse signal recovery in compressive sensing. For both applications, we provide precise conditions in the large-system limit for which AMPI achieves optimal performance. We furthermore use simulations to demonstrate that AMPI achieves near-optimal performance at low complexity in realistic, finite-dimensional systems.