The interaction between respiratory diseases of influenza and pneumonia is vital to understanding how coinfection might affect the transmission of pneumonia. We formulate an ordinary differential equation-based model to analyze the qualitative behavior of influenza and pneumonia coinfection. We determine the effective reproduction number and show that disease-free, boundary, and endemic equilibria are locally and globally stable in the absence of coinfection. Further, the model undergoes a backward bifurcation in the presence of the coinfection. Epidemiologically, this means that to eliminate disease, it may not be sufficient to bring the reproduction number below one. We then present numerical simulations to support the analytical results.