All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We are not aware of any verses or Ahadeeth about what these people claim that if the mat is not folded, the devil will pray on it.
You have to know that the devil is the worst disbeliever in Allaah and he is very far away from performing the prayer. He is called a devil because he is far removed from good and righteous actions, and very far from the Mercy of Allaah.
So the devil is the furthest creature from the prayer and the mention of Allaah.
Allaah Knows best.