Along with MITHRA and SRAOSHA, he judges the souls of the dead. We’re not sure what role the others play (do they split the workload between them?), but RASHNU himself provides the Golden Scales of Justice upon which the souls are weighed.
As the Angel of Final Judgment, we suspect that RASHNU is also standing by to sound the Last Trump and bring about the end of the world. But that’s just a wild guess.