TaSe3 is a layered van der Waals semimetal with several inverted band gaps throughout the entire Brillouin zone and nontrivial Z(2) topological indices, which place it at the boundary between a strong and a weak topological phase. Our transport experiments reveal a quadratic nonsaturating magnetoresistance (MR) with values reaching 10(4)% at 1.8 K and 14 T, whose origins have to be searched in the material's band structure. Here we combine angle-resolved photoelectron spectroscopy experiments, also with spin resolution, with ab initio calculations based on density functional theory in order to draw a connection between the Fermi surface topology and the measured transport properties. Simulations based on the calculated Fermi surface clarify that electron-hole compensation plays an important role for the observed MR in the bulk material. At the surface, the position of Fermi level differs, and it can be controlled by alkali metal deposition which accounts not only for the energy shift of the bands but it slightly modifies the dispersion of the valence and conduction bands. We propose that the observed band-gap renormalization might offer a route for engineering the topological phase in TaSe3, alternative to strain.