Refractory high entropy alloys (RHEAs) have shown great promise for a multitude of advanced engineering applications due to their high mechanical stability from cryogenic temperatures to 1600 °C. However, the low ductility they exhibit at room temperature limits their machinability when traditional machining techniques are used. Studies on the nontraditional machining of RHEAs, on the other hand, are very limited. In the present work, MoNbTaTiZr RHEAs subjected to wire electrical discharge machining (WEDM) were examined using electron microscopy, contact profilometry, and nanoindentation. Voids, microcracks, and recast material were observed on the machined surfaces. A transition from roughing to finishing decreased the amount of recast material, but the density of observable voids and microcracks on the surface increased. Optimal results were obtained by finishing, where the surface quality was improved by the removal of recast material remaining from prior passes. The brass wire electrode provided a smoother surface while the copper-core electrode provided minor heat-affected zone in the MoNbTaTiZr samples. In the roughing cuts, a 7.00% improvement in surface roughness was achieved using the copper-core electrode compared to the brass electrode. In the semi-finishing and finishing cuts, the brass wire electrode provided improvements of 13.68% and 22.68%, respectively, compared to the copper-core electrode. On the other hand, the wear of the brass electrode was as much as 20.98% higher than that of the copper-core electrode.