We study the Schwinger model at finite temperature and density using a variational algorithm for near-term quantum devices. We adapt $\beta$-VQE, a classical-quantum hybrid algorithm with a neural network, to evaluate thermal and quantum expectation values and study the phase diagram for the massless Schwinger model along with the temperature and density. By comparing the exact variational free energy, we find that the variational algorithm works for the Schwinger model for $T>0$ and $\mu$>0. As a result, we obtain a qualitative picture of the phase diagram for the massless Schwinger model.