Hydrogen can play a key role in the gradual transition towards a full decarbonization of the combustion sector, e.g., in power generation. Despite the advantages related to the use of this carbon-free fuel, there are still several challenging technical issues that must be addressed such as the thermoacoustic instability triggered by hydrogen. Given that burners are usually designed to work with methane or other fossil fuels, it is important to investigate their thermoacoustic behavior when fueled by hydrogen. In this framework, the present work aims to propose a methodology which combines Computational Fluid Dynamics CFD (3D Reynolds-Averaged Navier-Stokes (RANS)) and Finite Element Method (FEM) approaches in order to investigate the fluid dynamic and the thermoacoustic behavior introduced by hydrogen in a burner (a lab-scale bluff body stabilized burner) designed to work with methane. The case of CH4-air mixture was used for the validation against experimental results and benchmark CFD data available in the literature. Numerical results obtained from CFD simulations, namely thermofluidodynamic properties and flame characteristics (i.e., time delay and heat release rate) are used to evaluate the effects of the fuel change on the Flame Response Function to the acoustic perturbation by means of a FEM approach. As results, in the H2-air mixture case, the time delay decreases and heat release rate increases with respect to the CH4-air mixture. A study on the Rayleigh index was carried out in order to analyze the influence of H2-air mixture on thermoacoustic instability of the burner. Finally, an analysis of both frequency and growth rate (GR) on the first four modes was carried out by comparing the two mixtures. In the H2-air case the modes are prone to become more unstable with respect to the same modes of the case fueled by CH4-air, due to the change in flame topology and variation of the heat release rate and time delay fields.