Berry curvature in Weyl semimetals leads to intriguing magnetoconductivity and magnetothermal transport properties. Here, we explore the impact of the tilting of the Weyl nodes on the magnetoconductivity of type-I and type-II Weyl semimetals using the Berry-curvature-connected Boltzmann transport formalism. We find that in addition to the quadratic magnetic field (B) corrections induced by the tilt, there are also anisotropic and B-linear corrections in several elements of the conductivity matrix. For the case of magnetic field applied perpendicular to the tilt direction, we show the existence of previously unexplored B-linear transverse conductivity components. For the other case of magnetic field applied parallel to the tilt axis, the B-linear corrections appear in the longitudinal conductivity, giving rise to anisotropic magnetoresistance measurements. Our systematic analysis of the full magnetoconductivity matrix predicts several specific experimental signatures related to the tilting of the Weyl nodes in both type-I and type-II Weyl semimetals.