Reactive Rayleigh-Taylor systems are characterized by the competition between the growth of the instability and the rate of reaction between cold (heavy) and hot (light) phases. We present results from state-of-the-art numerical simulations performed at high resolution in 2d by means of a self-consistent lattice Boltzmann (LB) method which evolves the coupled momentum and temperature equations and includes a reactive term. We tune parameters in order to address the competition between turbulent mixing and reaction, ranging from slow- to fast-reaction rates. We also study the mutual feedback between turbulence evolution driven by the Rayleigh-Taylor instability and front propagation against gravitational acceleration. We quantify both the enhancement of "flame" propagation due to turbulent mixing for the case of slow reaction-rate as well as the slowing-down of turbulence growth for the fast-reaction case, when the front quickly burns the gravitationally unstable phase. An increase of intermittency at small scales for temperature characterizes the case of fast reaction, associated to the formation of sharp wrinkled fronts separating pure burnt/unburnt fluids regions.