We show that optical Tamm states (OTSs) with nonreciprocal dispersion can be formed at the boundary separating two different magnetophotonic crystals magnetized in the Voigt geometry. At the frequencies of the Tamm states, one-way optical tunneling can be achieved. The nonreciprocity features of OTSs originate from the simultaneous violation of reciprocity, time-reversal, and all related spatial symmetries in the system. Our predictions are confirmed by the nonreciprocal dispersion of interface modes, unidirectional transmission spectra, and field distributions for our system. Such theoretical results may provide a mechanism to create compact optical isolators.