We have estimated the energy deposition of runaway electrons into the tungsten/EUROFER blanket structure for reactor DEMO conditions and calculated the consequent level of thermal erosion. Our simulations indicate that the heat generated by runaway electrons may pose a major lifetime limitation for the W armor. We find that the minimum thickness of W necessary to prevent EUROFER from stress destruction at high temperatures, min, could be already too large for an efficient cooling. Tungsten layers of thickness min would erode by surface melting and vaporization since the thermal conductivity time is much larger than expected exposure time to runaways.