A modified lattice Boltzmann pseudopotential multicomponent and multiphase model is developed and analyzed. The model is capable of simulating large density ratios and viscosity ratios while ensuring thermodynamic consistency, and has a tunable surface tension. An improved source term is introduced to achieve the tunable surface tension with little influence on the thermodynamic consistency. The effect of surface tension on the bubble dissolution process under pressure is investigated. The results show that dramatic deformation and oscillation amplitudes result from larger surface tension, leading to a smaller equilibrium radius, consistent with Laplace's law. The convection process on the interface is the dominant factor in the oscillation of the average dissolved gas concentration. The dissolved gas concentration takes longer to reach equilibrium than the bubble radius due to the slow diffusion process. The average dissolved gas concentration increases with increasing initial pressure difference and surface tension, and the relationship between the average gas phase partial pressure and the average dissolved gas concentration is linear for different surface tensions, consistent with Henry's law. Although surface tension influences the dissolved gas concentration, it does not impact Henry's coefficient. The average dissolved gas concentration and bubble radius display a linear relationship with the bubble energy.