The main question of this paper is: What happens to the sparse (toric) resultant under vanishing coefficients? More precisely, let f_1, ..., f_n be sparse Laurent polynomials with supports A_1, ..., A_n and let Z_1 be a superset of A_1. Naturally a question arises: Is the sparse resultant of f_1, f_2, ..., f_n with respect to the supports Z_1, A_2, ..., A_n in any way related to the sparse resultant of f_1, f_2, ..., f_n with respect to the supports A_1, A_2, ..., A_n? The main contribution of this paper is to provide an answer. The answer is important for applications with perturbed data where very small coefficients arise as well as when one computes resultants with respect to some fixed supports, not necessarily the supports of the f_i's, in order to speed up computations. This work extends some work by Sturmfels on sparse resultant under vanishing coefficients. We also state a corollary on the sparse resultant under powering of variables which generalizes a theorem for Dixon resultant by Kapur and Saxena. We also state a lemma of independent interest generalizing Pedersen's and Sturmfels' Poisson-type product formula.