Is there any weak or fabricated hadith to suggest that Soorah Alam Nashra has any healing effect?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

We could not find any Marfoo‘ (directly attributed to the Prophet, sallallahu ‘alayhi wa sallam) hadeeth regarding this chapter of the Quran except what Ath-Tha'labi and Al-Waahidi narrated in their Tafseer (exegesis) books and Al-Mustaghfiri in his book Fadhaa'il Al-Quran, with the following wording, "Whoever recites Alam nashrah laka sadrak, it is as if he came to me while I am distressed and relieved my distress." However, this narration is not authentic. Al-Khateeb Ash-Shirbeeni said, "It is a fabricated hadeeth." [As-Siraaj Al-Muneer]

Shaykh Muhammad Al-Ameen Al-Harari said, "It has no basis." [Tafseer Haada'iq Ar-Rawh wa Ar-Rayhaan]

Al-Fayroozabaadi said about the virtue of this chapter, "There are two weak reports narrated in this regard. The first reads, 'Whoever recites it, it is as if he came to me while I am distressed and relieved my distress.' The second report reads, 'O ‘Ali! Whoever recites it, it is as if he has fed the poor of my nation, and he shall be rewarded with a beautiful gown for each verse he read on the Day of Gathering.'" [Basaa'ir Thawi At-Tamyeez]

As for the stories about it, Al-Qaadhi At-Tanookhi said:

"As for the report about Soorah Ash-Sharh, Abu Bakr ibn Shujaa‘, the Baghdadi reciter of the Quran, who was my successor at the Mint in the marketplace of Al-Ahwaaz in 346 AH, he was the keeper of the mosque there and was a trustworthy Hadeeth scholar and one of the trustees of Al-Qaadhi Al-Ahnaf. His name was Muhammad ibn ‘Abdullaah ibn `Ali ibn Muhammad ibn Abu Ash-Shawaarib. He related to us the following report with its chain of narration, but I cannot recall it. As I remember, the text of the report was as follows, 'A pious man once got overwhelmed with distress and faced many difficulties and hardships (to such an extent) that he was at the verge of despair. He was once walking and composed a pessimistic poetic verse about death being better for the humiliated than life, he heard a voice but could not see the speaker, reciting optimistic poetic verses urging him to recite Soorah Ash-Sharh whenever overwhelmed by distress. He said, 'I continued to recite it in the prayer and my distress was relieved, my affairs were made easy and facilitated.' Other than him also related this story to me.'" [Al-Faraj ba‘d Ash-Shiddah]

In any case, there is no harm in reciting any chapter of the Quran with the purpose of making Tawassul (coming nearer) to Allah through it and imploring Him to grant the reciter his lawful needs. ‘Imraan bin Husayn, may Allah be pleased with him, narrated that the Prophet, sallallahu ‘alayhi wa sallam, said, "Whoever recites the Quran, then let him ask Allah by it; for indeed there will come a people who will recite the Quran, asking people by it." [Ahmad - Al-Albaani graded it saheeh (sound)]

‘Imraan ibn Husayn also narrated that he once passed by a reciter who was reciting Quran and then started begging for money. So, he (‘Imraan) said, "Indeed we are from Allah and indeed to Him we shall return." Then he said, "I heard the Messenger of Allah, sallallahu ‘alayhi wa sallam, say, "Whoever recites the Quran, then let him ask Allah by it; for indeed there will come a people who will recite the Quran, asking people by it." [At-Tirmithi graded it hasan (good) and Al-Albaani graded it saheeh, in his book Saheeh At-Targheeb]

Al-Mubaarakfoori said, "Asking Allah by it may mean that the person implores Allah by the Quran for anything in this worldly life or the Hereafter. It may also mean that he implores Allah for His mercy whenever he recites a verse about His mercy, or that he recites the reported supplications after reciting the Quran." [Tuhfat Al-Ahwathi]

Allah knows best.