We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. Using the sign function Newton iteration as the solver, we show that reduced precisions, such as the half precision, can be used as the solver precision, say, $u_s$ to accelerate the solution of Lyapunov equations of condition number up to $1/u_s$ without compromising its quality, especially in memory-efficient systems where the communication cost constitutes a negligible part of the overall runtime.