Is it permissible for a male to treat a female patient using ruqyah syar’iyyah and touching the patient lined with a cloth?
The ruling of a man touching a non-mahram woman is haram (prohibited). In a hadith narrated from Ma’qil bin Yasar RA, he said the Messenger PBUH said:
لأَنْ يُطْعَنَ فِي رَأْسِ أَحَدِكُمْ بِمِخْيَطِ مِنْ حَدِيدٍ خَيرٌ لَهُ مِنْ أَنْ يَمَسَّ امْرَأةً لَا تَحِلُّ لَهُ
“If anyone of you is pierced with a steel needle, certainly that is better for him compared to him touching a prohibited woman for him (non-mahram).” 
Although the literal meaning showed the prohibition on man, however, the ruling is also applicable for the consequence as explained by Mulla ‘Ali al-Qari:
الْخِطَاب إِذَا وَرَدَ بِلَفْظِ الذُّكُورِ كَانَ خِطَابًا لِلنِّسَاءِ إِلَّا فِي مَوَاضِعَ مَخْصُوصَةٍ
“A particular khitab (sentence or statement) when there is a pronouncement stating it is for men, concurrently, the khitab is also applicable for women, except in certain situations.” 
I’m reminded of a story of Syeikh Hasan Habannakah when a woman wanted to shake hands with him and refused. Following the incident, it is said that the Syeikh may be in a pure state of ablution at the time, the Syeikh then responded: “No. However, the truth of the matter is Rasullullah PBUH never touched the hands of any non-mahram woman.”
This is based on a hadith from Saidatina A’ishah radhiyallahu ‘anha:
وَاللَّهِ مَا مَسَّتْ يَدُهُ يَدَ امْرَأَةٍ قَطُّ فِي الْمُبَايَعَةِ، مَا يُبَايِعُهُنَّ إِلاَّ بِقَوْلِهِ
“By Allah, his hand (the Prophet PBUH’s) never touched any lady during that pledge of allegiance. He did not receive their pledge except by speech,” 
Thus, it is best to avoid any contact (touching), but just use recitation for the patient if it is a non-mahram male and female. The important matter in the treatment is the recitation and not the touching. Wallahua’lam.
 Narrated by Al-Thabarani, Mu’jam Al-Kabir, 20/211
 Mirqat Al-Mafatih, 2/423.
 Narrated by Imam Bukhari, no. 4891