Two-phase flows combined with heat and mass transfer may present configurations in which dilatable gas pockets coexist along with nearly-incompressible liquid. Finite Volume discretisation of such compressible equations are known to introduce excessive numerical diffusion and to severely restrain time steps. Adequate solver has been developed to tackle these issues (Chalons, 2016), especially in the presence of a sharp interface (Zou, 2020, 2022). To handle multi scales inherently arising in complex two-phase flows, adaptive meshing is an efficient strategy to reduce computation cost compared to uniform grids. In the present study, we use the Multi-Resolution Analysis (MRA) (Harten, 1994), based on local wavelet basis decomposition. This method provide a measure of local error to select regions to refine or derefine and garantees a precise control of the additionnal error introduced by mesh operations, independantly of conservation equations and problems. This approach was successfully applied to two-phase flows with sharp interface representation (Han, 2014) and we extend it to low-Mach flows. We focus on configurations with heat and mass transfer, to exhibit accuracy and advantages of the MRA method, with respect to other adaptive methods (mainly Adaptive Mesh Refinement (AMR) (Berger, 1989)).