The progression of viral infection within the human body is governed by a complex interplay between the pathogen and the immune response. The initial phase of the innate immune response is driven by inflammatory cytokines and interferons produced by infected target cells and tissue-resident macrophages. These inflammatory cytokines not only amplify the immune response but also initiate programmed cell death, which helps slow the spread of the infection. The propagation of the infection within tissues can be modeled as a reaction-diffusion wave, where the speed of this wave is linked to the virus virulence, and the overall viral load determines its infectivity. In this study, we demonstrate that inflammation reduces both the speed and viral load of the infection wave, and we establish the conditions necessary to halt the spread of the infection. Depending on the relative strength of the infection and the immune response, there are three possible outcomes of infection progression. If the virus replication number is sufficiently low, the infection does not develop. For intermediate values of this parameter, the infection spreads within the affected tissue at a decreasing speed and amplitude before ultimately being eliminated. However, if the virus replication number is high, the infection propagates as a reaction-diffusion wave with a constant speed and amplitude. These findings are derived using analytical methods and are corroborated by numerical simulations. Additionally, we explore viral diffusion, comparing the conventional parabolic diffusion model with the hyperbolic diffusion model, which is introduced to address the limitation of infinite propagation speed. Our results show that while the viral load remains the same across both models, the wave speed in the hyperbolic model is smaller and approaches that of the parabolic model as the relaxation time decreases.