Is there any evidence that the Prophet, peace be upon him, heals the sick through dreams, as some claim they were cured after he touched them in their sleep, especially since one of these patients is a memorizer of the Qur'an despite being illiterate?

There is no evidence from the Quran or Sunnah that a sick person is cured by seeing the Prophet (peace be upon him) in a dream. However, Allah may heal the sick, and the dream may have a positive psychological effect, thus becoming a cause for their recovery. Seeing the Prophet (peace be upon him) in his true form is a genuine vision and one of the glad tidings.