Many cellular populations cooperate through the secretion of diffusible extracellular resources, such as digestive enzymes or virulence factors. Diffusion of these resources leads to long-range intercellular interactions, creating the possibility of cooperation but also the risk of exploitation by non-producing neighbors. In the past, considerable attention has been given to game-theoretic lattice models of intercellular cooperation. In these models, coexistence is commonly observed between cooperators (corresponding to resource producers) and cheaters (corresponding to nonproducers). However, these models consider only interactions between direct competitors. We find that when individuals are allowed to interact non-locally through the diffusion of a shared resource coexistence between cooperators and cheaters is lost. Instead, we find population dynamics similar to simple competition, either neutral or biased, with no balancing selection that would favor coexistence. Our results highlight the importance of an accurate treatment of diffusion of shared resources and argue against the generality of the conclusions of game-theoretic lattice models.