This study developed hydrogels containing different ratios of TA using polyvinyl alcohol (PVA) and polyethyleneimine (PEI) polymers crosslinked with tannic acid (TA) for the treatment of burn wounds. Various tests, such as scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), swelling, moisture retention, contact angle, tensile strength, the scratch test, antibacterial activity and the in vitro drug-release test, were applied to characterize the developed hydrogels. Additionally, the hydrogels were examined for cytotoxic properties and cell viability with the WST-1 test. TA improved both the self-healing properties of the hydrogels and showed antibacterial activity, while the added gentamicin (GEN) further increased the antibacterial activities of the hydrogels. The hydrogels exhibited good hydrophilic properties and high swelling capacity, moisture retention, and excellent antibacterial activity, especially to S. aureus. In addition, the swelling and drug-release kinetics of hydrogels were investigated, and while swelling of hydrogels obeyed the pseudo-second-order modeling, the drug release occurred in a diffusion-controlled manner according to the Higuchi and Korsmeyer–Peppas models. These results show that PVA/PEI-based hydrogels have promising potential for wound dressings with increased mechanical strength, swelling, moisture retention, self-healing, and antibacterial properties.