There is nothing in the Qur'an or authentic hadiths that mentions anything about Iblees praying on an unfolded prayer mat. There is also nothing in the Qur'an or authentic hadiths that refers to Iblees, or any other shaytan, praying in the first place.
From a fiqh point of view, folding the prayer mat, partially folding it, or leaving it unfolded is permitted. The only concern is keeping a prayer mat free of najas. Other than that, you may elect either option.
Just for the sake of mentioning it, there is an athar (not a hadith attributed to the Prophet) about a female jinn who used to visit the Prophet, but was late one day as she had to go to a girl's (another jinn) funeral in India. On her way, she saw a shaytan praying. This athar was mentioned in "The Fabricated Pearls in Fabricated Hadiths" (اللآلىء المصنوعة في الأحاديث الموضوعة) by Abdul-Rahman ibn Abi Bakr and Jalal Al-Din Al-Suyuti, which needless to say is fabricated, and definitely not authentic.