The neuronal microtubule-associated protein tau, MAPT, is central to the pathogenesis of many dementias. Autosomal-dominant mutations in MAPT cause inherited frontotemporal dementia (FTD), but the underlying pathogenic mechanisms are unclear. Using human stem cell models of FTD due to MAPT mutations, we find that tau becomes hyperphosphorylated and mislocalizes to cell bodies and dendrites in cortical neurons, recapitulating a key early event in FTD. Mislocalized tau in the cell body leads to abnormal microtubule movements in FTD-MAPT neurons that grossly deform the nuclear membrane. This results in defective nucleocytoplasmic transport, which is corrected by microtubule depolymerization. Neurons in the post-mortem human FTD-MAPT cortex have a high incidence of nuclear invaginations, indicating that tau-mediated nuclear membrane dysfunction is an important pathogenic process in FTD. Defects in nucleocytoplasmic transport in FTD point to important commonalities in the pathogenic mechanisms of tau-mediated dementias and ALS-FTD due to TDP-43 and C9orf72 mutations.