Praise be to Allah.
There is nothing wrong with reading the Quran from the Mus-haf during a Sunnah prayer such as Qiyam Al-Layl .
But in the case of obligatory prayers, it is disliked to do that, because in most cases there is no need for it. But if there is a need, then there is nothing wrong with reading from the Mus-haf in that case.
Ibn Qudamah (may Allah have mercy on him) said in Al-Mughni, 1/335:
“Ahmad said: There is nothing wrong in leading the people in praying Qiyam whilst looking at the Mus-haf. It was said to him: What about obligatory prayers? He said: I have not heard anything concerning that. Al-Qadi said: It is disliked in obligatory prayers, but there is nothing wrong with it in voluntary prayers if one has not memorised (Quran), but if one has memorised Quran then it is also disliked. He said: Ahmad was asked about leading the prayers whilst reading from the Mus-haf in Ramadan. He said: If that is necessary (it may be done)… and it was narrated from Ibn Hamid that it is equally permissible in both Nafl and obligatory prayers.
The evidence for it being permissible is the report narrated by Abu Bakr Al-Athram and Ibn Abu Dawud with their Isnad (chain of narration) from `Aishah (may Allah be pleased with her), according to which she would be led in prayer by a slave of hers who read from the Mus-haf .
Al-Zuhri (may Allah have mercy on him) was asked about a man who read from the Mus-haf in Ramadan. He said: The best ones among us used to read from the Mus-hafs…
Reading from the Mus-haf has been permitted because of the need to listen to the Quran and recite it in night prayers (Qiyam).
The ruling on it being disliked applies only to those who have memorised the Quran, because they will be needlessly distracted from proper focus in prayer (Khushu`) by that, and from looking at the place of prostration. And it is disliked in obligatory prayers in general, because usually there is no need for it.”
An-Nawawi (may Allah have mercy on him) said in Al-Majmu` (4/27):
“If a person reads Quran from the Mus-haf, this does not invalidate his prayer, whether he has memorised the Quran or not; in fact that is obligatory if he has not memorised Al-Fatihah.
What we have mentioned, that reading from the Mus-haf does not invalidate the prayer, is our view and the view of Malik, Abu Yusuf, Muhammad and Ahmad.”
Shaykh Ibn Baz (may Allah have mercy on him) was asked: Is it permissible for the Imam to read from the Mus-haf during the five daily prayers, especially Fajr when lengthy reading is required and there is the fear of making mistakes or forgetting?
“That is permissible if there is a need for it, just as it is permissible to read from the Mus-haf in Tarawih for one who has not memorised the Quran. Dhakwan, the freed slave of `Aishah (may Allah be pleased with her) used to lead her in prayer in Ramadan, reading from the Mus-haf, as was narrated by Al-Bukhari. It is Sunnah to recite at length in Fajr prayer, so if the Imam has not memorised Al-Mufassal or anything else from the rest of the Quran, it is permissible for him to recite from the Mus-haf. But it is prescribed for him to strive to memorise the Quran, or at least to memorise Al-Mufassal, so that he will not need to read from the Mus-haf. Al-Mufassal refers to the portion of the Quran that starts with Surah Qaf, up to the end of the Quran. Whoever strives to memorise, Allah will make it easy for him, as He says (interpretation of the meaning):
“And whosoever fears Allah and keeps his duty to Him, He will make a way for him to get out (from every difficulty)” [At-Talaq 65:2]
“And We have indeed made the Quran easy to understand and remember; then is there any one who will remember (or receive admonition)?” [Al-Qamar 54:17]
And Allah is the Source of strength."(Majmu` Fatawa Ibn Baz, 11/117)
And Allah knows best.