We consider the computation of the surface tension of the fluid-fluid interface for the Widom-Rowlinson [J. Chem. Phys. 52, 1670 (1970)] binary mixture from direct simulation of the inhomogeneous system. We make use of the standard mechanical route, in which the surface tension follows from the computation of the normal and tangential components of the pressure tensor of the system. In addition to the usual approach, which involves simulations of the inhomogeneous system in the canonical ensemble, we also consider the computation of the surface tension in an ensemble where the pressure perpendicular (normal) to the planar interface is kept fixed. Both approaches are seen to provide consistent values of the interfacial tension. The issue of the system-size dependence of the surface tension is addressed. In addition, simulations of the fluid-fluid coexistence properties of the mixture are performed in the semigrand canonical ensemble. Our results are compared with existing data of the Widom-Rowlinson mixture and are also examined in the light of the vapor-liquid equilibrium of the thermodynamically equivalent one-component penetrable sphere model.