Here we investigate, using relativistic fluid theory and Vlasov-Maxwell simulations, the local heating of a dense plasma by two crossing electron beams. Heating occurs as an instability of the electron beams drives Langmuir waves, which couple nonlinearly into damped ion-acoustic waves. Simulations show a factor 2.8 increase in electron kinetic energy with a coupling efficiency of 18%. Our results support applications to the production of warm dense matter and as a driver for inertial fusion plasmas.