Post-processing heat treatments (HT) are performed on Ti-6Al-4V components fabricated by Laser Powder Bed Fusion (L-PBF) to release the residual stresses and transform the non-equilibrium martensite into a dual-phase α+β microstructure. In the present study, post-processing sub-transus HT are conducted at various annealing temperatures for 2 h, with slow cooling either in a larger volumetric horizontal industrial furnace or a smaller volumetric tubular laboratory furnace. These two different furnaces were chosen to significantly modify the atmosphere and generate an oxygen-affected layer on the surface. The results show that the higher the annealing temperature, the coarser the α laths and the higher the β phase fraction. An oxygen-enriched hard and brittle alpha-case layer was revealed only for HT performed in the industrial furnace. The provided data allowed modeling of the oxygen diffusion in Ti-6Al-4V made by L-PBF. It has been found that without the alpha-case, post-processing HT helps to balance the strength/ductility compromise of the alloy in relation to the microstructural evolution. On the other hand, when an alpha-case layer greater than 50 µm is present, the elongation to failure is impacted and decreases as the alpha-case depth increases. Meanwhile, the impact toughness is less affected and remains mainly microstructure-dependent. The loss in ductility was attributed to the presence of cracks that develop at the surface of the samples during tensile solicitation. The opening rate of the crack lips is proportional to the applied strain, and the cracks spread through the brittle alpha-case layer. It has been shown for the first time that the cracks are visible on the sample’s surface for stress values lower than half of the material’s yield stress. The post-processing HT at 800 °C for 2 h provided a strong and ductile Ti-6Al-4V, which showed a minor effect of the alpha-case on the tensile properties.