As we proved earlier, for any triangulated category C_ endowed with a weight structure w and a triangulated subcategory D_ of C_ (strongly) generated by cones of a set of morphism S in the heart Hw_ of w there exists a weight structure w' on the Verdier quotient C'_=C_/D_ such that the localization functor C_→C'_ is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of C'_ of non-negative (resp. non-positive) weights there exists its preimage in C_ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if D_ is generated by objects of Hw_ then any object of Hw'_ lifts to Hw_. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.