We investigate the numerical solution of Poisson equations on dynamically adapted structured grids generated by multiresolution analysis, arising in the numerical simulation of time-dependent problems modeled by evolutionary partial differential equations (PDEs) disclosing propagating fronts such as streamer discharge phenomena. A multiresolution technique not only involves important savings in computational costs, but we also prove that it guarantees numerical approximations to the Poisson equation within an accuracy tolerance. In contrast to most adaptive meshing approaches in the literature that solve such elliptic PDEs level-wise and thus at uniform resolution throughout the set of adapted grids, we introduce a numerical procedure mainly based on inter-level operations to represent the elliptic operators on the adapted grid. In this way the discrete Poisson equation can be solved at once over the entire computational domain as a completely separate process, strongly coupling inter-grid relations and independently of the mesh generation or any other grid-related data structure or geometric consideration. We investigate the validity of both the theoretical characterization as well as the numerical construction of the discrete operators in the context of streamer discharge simulations. Multiresolution error bounds are thus evaluated in a simpler configuration with analytical solution that nevertheless mimics the spatial structure found in such physical configurations. The complete model is then considered to simulate double-headed streamers with photoionization mechanisms, for which we study the performance and capabilities of various direct and iterative linear solvers. The numerical strategy is finally assessed by showing its ability to accurately describe the complex interaction dynamics of two positive streamer discharges, a problem especially relevant for applications.