We propose and analyze a conforming finite element method for a two-dimensional nonisothermal fluid-membrane interaction problem. The problem consists of a Navier-Stokes/heat system, commonly known as the Boussinesq system, in the free-fluid region, and a Darcy-heat coupled system in the membrane. These systems are coupled through buoyancy terms and a set of transmission conditions on the fluid-membrane interface, including mass conservation, balance of normal forces, the Beavers-Joseph-Saffman law, and continuity of heat flux and fluid temperature. We consider the standard velocity-pressure-temperature variational formulation for the Boussinesq system, along with a dual-mixed scheme coupled with a primal formulation for the Darcy and Heat equations in the membrane region. The latter yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. For the associated Galerkin scheme, we employ Bernardi-Raugel and Raviart-Thomas elements for velocities, piecewise constant elements for pressures, continuous piecewise linear functions for temperatures, and continuous piecewise linear functions for the Lagrange multiplier on a partition of the interface. We prove well-posedness for both the continuous and discrete schemes and derive corresponding error estimates. Finally, we present numerical examples to confirm the predicted convergence rates and demonstrate the performance of the method.