We compute the structural and electronic properties of ⟨111⟩-oriented $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{P}$ nanowire heterostructures using Keating's valence force field and a tight-binding model. We focus on the optical properties (exciton energies and polarization) of InAs quantum dots embedded in InP nanowires and on the height of InP and InAsP tunnel barriers embedded in InAs nanowires. We show that InAs quantum dots exhibit bright optical transitions, at variance with the highly mismatched $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ nanowire heterostructures. The polarization of the photons is perpendicular to the nanowire for thin InAs layers but rotates parallel to the nanowire for thick enough ones, as a result of the increasing light-hole character of the exciton. As for tunnel barriers, we show that the residual strains can significantly reduce the conduction band discontinuity in thin InAsP layers. This must be taken into account in the design of nanowire tunneling devices.