When integrated into microfluidic chips, ion-selective nanoporous polymer and solid-state membranes can be used for on-chip pumping, pH actuation, analyte concentration, molecular separation, reactive mixing, and molecular sensing. They offer numerous functionalities and are hence superior to paper-based devices for point-of-care biochips, with only slightly more investment in fabrication and material costs required. In this review, we first discuss the fundamentals of several nonequilibrium ion current phenomena associated with ion-selective membranes, many of them revealed by studies with fabricated single nanochannels/nanopores. We then focus on how the plethora of phenomena has been applied for transport, separation, concentration, and detection of biomolecules on biochips.