There is nothing wrong with reading Qur’aan from the Mus-haf during a naafil prayer, such as qiyaam al-layl.
But in the case of obligatory prayers, it is makrooh 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 Qudaamah (may Allaah have mercy on him) said in al-Mughni, 1/335:
Ahmad said: There is nothing wrong will leading the people in praying qiyaam 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-Qaadi said: It is makrooh in obligatory prayers, but there is nothing wrong with it in voluntary prayers if one has not memorized (Qur’aan), but if one has memorized Qur’aan then it is also makrooh. He said: Ahmad was asked about leading the prayers whilst reading from the Mus-haf in Ramadaan. He said: If that is necessary (it may be done)… and it was narrated from Ibn Haamid that it is equally permissible in both naafil and obligatory prayers.
The evidence for it being permissible is the report narrated by Abu Bakr al-Athram and Ibn Abi Dawood with their isnaads from ‘Aa’ishah, according to which she would be led in prayer by a slave of hers who read from the Mus-haf.
Al-Zuhri was asked about a man who read from the Mus-haf in Ramadaan. 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 Qur’aan and recite it in night prayers (qiyaam).
The ruling on it being makrooh applies only to those who have memorized Qur’aan, because they will be needlessly distracted from proper focus in prayer (khushoo’) by that, and from looking at the place of prostration. And it is makrooh in obligatory prayers in general, because usually there is no need for it.
Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’ (4/27):
If a person reads Qur’aan from the Mus-haf, this does not invalidate his prayer, whether he has memorized the Qur’aan or not; in fact that is obligatory if he has not memorized al-Faatihah.
What we have mentioned, that reading from the Mus-haf does not invalidate the prayer, is our view and the view of Maalik, Abu Yoosuf, Muhammad and Ahmad.
Shaykh Ibn Baaz (may Allaah 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 Taraweeh for one who has not memorized the Qur’aan. Dhakwaan, the freed slave of ‘Aa’ishah (may Allaah be pleased with her) used to lead her in prayer in Ramadaan, reading from the Mus-haf, as was narrated by al-Bukhaari in his Saheeh in a mu’allaq majzoom report. It is Sunnah to recite at length in Fajr prayer, so if the imam has not memorized al-Mufassal or anything else from the rest of the Qur’aan, it is permissible for him to recite from the Mus-haf. But it is prescribed for him to strive to memorize the Qur’aan, or at least to memorize al-Mufassal, so that he will not need to read from the Mus-haf. Al-Mufassal refers to the portion of the Qur’aan that starts with Soorat Qaf, up to the end of the Qur’aan. Whoever strives to memorize, Allaah will make it easy for him, as He says (interpretation of the meaning):
“And whosoever fears Allaah and keeps his duty to Him, He will make a way for him to get out (from every difficulty)”
“And We have indeed made the Qur’aan easy to understand and remember; then is there any one who will remember (or receive admonition)?”
And Allaah is the Source of strength.
Majmoo’ Fataawa Ibn Baaz, 11/117
And Allaah knows best.