Sparse inverse problems in the presence of Poisson noise with physical constraints arise in a variety of applications, including photon-limited imaging systems based on compressed sensing. The performance of compressed sensing in these settings can be markedly different from classical settings. Prior results on Poisson compressed sensing provided upper bounds on mean squared error performance; however, it was unknown whether those bounds were tight or if other estimators could achieve significantly better performance. This work provides minimax lower bounds on mean-squared error for sparse Poisson inverse problems under physical constraints. The lower bounds are complemented by minimax upper bounds which match the lower bounds for certain problem sizes and noise levels. The upper and lower bounds reveal several distinctions from the classical compressed sensing setup due to the interplay between the Poisson noise model, the sparsity of the signal, and the physical constraints. For example, error decay rates depend heavily upon the sparsifying basis of the signal. In addition, in many application-relevant scenarios signal acquisition via simple downsampling can significantly outperform compressed sensing. This is joint work with Xin Jiang and Garvesh Raskutti.