The notion of Khovanskii bases was introduced by Kaveh and Manon. It is a generalization of the notion of SAGBI bases for a subalgebra of polynomials. The notion of SAGBI bases was introduced by Robbiano and Sweedler as an analogue of Gröbner bases in the context of subalgebras. A Hibi ideal is an ideal of a polynomial ring that arises from a distributive lattice. For the development of an analogy of the theory of Hibi ideals and Gröbner bases within the framework of subalgebras, in this paper, we investigate when the set of the polynomials associated with a distributive lattice forms a Khovanskii basis of the subalgebras it generates. We characterize such distributive lattices and their underlying posets. In particular, generalized snake posets and $\{(2+2),(1+1+1)\}$-free posets appear as the characterization.