The bias dependence of the tunnel magnetoresistance ͑TMR͒ of Fe/MgO/Fe tunnel junctions is investigated theoretically with a fully self-consistent scheme that combines the nonequilibrium Green's-function method with density-functional theory. At voltages smaller than 20 mV the I-V characteristics and the TMR are dominated by resonant transport through narrow interface states in the minority-spin band. In the parallel configuration this contribution is quenched by a voltage comparable to the energy width of the interface state, whereas it persists at all voltages in the antiparallel configuration. At higher bias the transport is mainly determined by the relative positions of the ⌬ 1 band edges in the two Fe electrodes, which causes a decrease in the TMR.