Induced pluripotent stem cells (iPSCs), and cells derived from them, have become key tools to model biological processes and disease mechanisms, particularly in cell types such as neurons that are difficult to access from living donors. Here, we present the first map of regulatory variants in an iPSC-derived cell type. To investigate genetic contributions to human sensory function, we performed 123 differentiations of iPSCs from 103 unique donors to a sensory neuronal fate, and measured gene expression, chromatin accessibility, and neuronal excitability. Compared with primary dorsal root ganglion, where sensory nerves collect near the spinal cord, gene expression was more variable across iPSC-derived neuronal cultures, particularly in genes related to differentiation and nervous system development. Single cell RNA-sequencing revealed that although the majority of cells are neuronal and express the expected marker genes, a substantial fraction have a fibroblast-like expression profile. By applying an allele-specific method we identify 3,778 quantitative trait loci influencing gene expression, 6,318 for chromatin accessibility, and 2,097 for RNA splicing at FDR 10%. A number of these overlap with common disease associations, and suggest candidate causal variants and target genes. These include known causal variants at SNCA for Parkinson's disease and TNFRSF1A for multiple sclerosis, as well as new candidates for migraine, Parkinson's disease, and schizophrenia.