Why doesn't Mara suffer? Everyone born in this world is bound to suffer. 
Mara tempted Buddha so many times to abandon his quest which he did not. Yet Mara herself never suffered due to her own Karma or due to her belief system. Mara is alive even today.

My question is why not Mara, the tempter, suffers or suffered due to her belief system and actions ?
Mara (a psychic that decides to harass Buddhists) does suffer according to the suttas: 


  Then Mara the Evil One, realizing, "The bhikkhuni Vajira knows me," sad and disappointed, disappeared right there. SN 5.10
  
  
  
  ‘Get out, Evil One, Evil One, get out. Do not annoy a Tathāgata or a Tathāgata's disciple, lest for a long time there be woe and sorrow for
  you.’...While he was looking around, Evil One, Dūsin the Māra deceased from that place and arose in the Great Niraya Hell. What was that Niraya
  Hell like where Dūsin the Māra was boiled, For striking the disciple Vidhura
  and the brahman Kakusandha? It was that of the hundred iron spikes,
  all suffered separately. This was the Niraya Hell where Dūsin was
  boiled, For striking the disciple Vidhura and the brahman Kakusandha. MN 50