Davydov-Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. We consider such deformations of finite tensor categories and exact tensor functors between them. In arXiv:1910.06094, DY cohomology with coefficients was introduced and related to the comonad cohomology for a certain adjunction; in the case of tensor categories it reduces to the adjunction for the forgetful functor of the Drinfeld center. We first prove that the DY cohomology groups are isomorphic to the relative Ext groups for this adjunction. From this, we derive the following main results: the vanishing of the first DY cohomology group, long exact sequences of DY cohomology groups which allow to express the groups in terms of Hom spaces, and the existence of a Yoneda-type product on DY cocycles. We apply these results to the category of finite-dimensional modules over a finite-dimensional Hopf algebra and provide a method to compute explicit DY cocycles. We study in detail the examples of the bosonization of exterior algebras $\Lambda\mathbb{C}^k \rtimes \mathbb{C}[\mathbb{Z}_2]$, the Taft algebras and the restricted quantum group of $\mathfrak{sl}_2$ at a fourth root of unity $\bar U_{\mathbf{i}}(\mathfrak{sl}_2)$.