Benefit: The Ruling on Reading the Qurʾān from the Mus‘haf during Tarāwīḥ
The Answer: For the one who is incapable of reading from memory, then there is nothing wrong if he reads from the Mus-haf during the Tarāwīḥ prayer. This is because he will read the entire Qurʾān from its beginning to its end. So, if he isn’t a memorizer of the Qurʾān, then he may read from the Mus-haf. Some of the Salaf (righteous predecessors) used to do this.
Member of the Permanent Committee for Fatāwá, al-ʿAllāmah Shaykh Ṣāliḥ ibn Fawzān al-Fawzān, hafiẓahullāh
The Question: What about reciting the Qurʾān from the Mus-haf (during prayer) Noble Shaykh?
The Answer: There is no objection in that - that which is correct is that there is no objection in that. So if he needed to recite from the Mus-haf because he hasn’t memorized the Qurʾān, then there is nothing wrong if he reads from the Mus-haf. It has been established that ‘Aa’ishah’s (raḍīAllāhu ʿanhā) slave, Dhakwān, used to lead her in prayer (in Ramaḍān), reading from the Mus-haf as mentioned by Bukhārī (raḥimahullāh) in a mu’allaq report.
Also, the original principle is that there is nothing wrong with this, and whoever forbids this, then he should bring forth evidence. The original principle has permitted recitation from memory and recitation from the Mus-haf. This is the original principle, and whoever says, “It is forbidden [to read] from the Mus-haf," then he should bring forth evidence, and the original principle is that there is no evidence. So he has forbidden the original principle, and it is the permissibility of reciting the Qurʾān from the Mus-haf. The mother of the believers has implemented this, and she is from amongst the most knowledgeable of people (raḍīAllāhu ʿanhā).
Al-'Allāmah Shaykh ibn Bāz: http://www.ibnbaz.org.sa/mat/20173
The noble Shaykh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn (raḥimahullāh) was asked: There is a mosque whose congregation is few (in number) and there isn’t anyone from (the congregation) who is capable of reciting from memory. So, can the one who wants to lead the congregation, read with the Mus-haf by looking at it during the prayer – the night prayer in Ramaḍān, for example?
The noble Shaykh responded by saying:Naʿam (Yes). If the Imām hasn’t memorized anything from the Qurʾān, then this action is considered permissible for him (to do) – that he reads with the Mus-haf and there is no prohibition upon him regarding that, because this action is (implemented) for the sake of finishing the prayer, and the action (implemented) to finish the prayer does not harm, and Allāh is the granter of success.
Source: Hasheeyat Majmu’ Fatāwá Shaykh (raḥimahullāh)
Source for all fatāwá: http://www.sahab.net/forums/showthread.php?t=380892