Praise be to Allah.
This question covers a number of issues
1 – When is a Muslim not allowed to read the Qur’aan?
It is permissible for a Muslim to read Qur’aan in all situations, except when he is junub (in a state of major impurity following sexual activity); he is not permitted to read Qur’aan in this situation. Shaykh al-Islam [Ibn Taymiyah] said: “The scholars are unanimously agreed that it is forbidden for the person who is junub to read Qur’aan.” (See Tawdeeh al-Ahkaam by ‘Abd-Allaah al-Bassaam, vol. 1, p. 309)
With regard to what is mentioned about reading Qur’aan being dhikr, this is correct, indeed it is one of the best forms of dhikr, because it is the words of Allaah.
2 – The Prophet (peace and blessings of Allaah be upon him) forbade prayer at certain times. See the answer to Question # 8818. It was narrated that it is not allowed to pray or to bury the dead at those times, but that does not include reading Qur’aan. Rather it was narrated from the Prophet (peace and blessings of Allaah be upon him) that it is mustahabb to remember Allaah (dhikr) after Fajr, which undoubtedly includes reading Qur’aan. Al-Tirmidhi narrated that the Prophet (peace and blessings of Allaah be upon him) said: “Whoever prays Fajr in jamaa’ah, then sits and remembers Allaah until the sun comes up, then prays two rak’ahs, will have a reward like that of Hajj and ‘Umrah.” He said: the Messenger of Allaah (peace and blessings of Allaah be upon him) said, “Complete, complete, complete.” (al-Jumu’ah 535; classed as hasan by al-Albaani in Saheeh Sunan al-Tirmidhi, 480). Disallowing this is not valid unless there is evidence (daleel) to that effect; how can that be when there is evidence to indicate that it is mustahabb?
3 – Is sujood al-tilaawah (prostration during recitation of Qur’aan) considered to be salaah (prayer)?
Shaykh Ibn ‘Uthaymeen said:
Some of the scholars were of the view that it is prayer.
Some of the scholars were of the view that it is not prayer, because the definition of prayer does not apply to it, as it is not proven in the Sunnah that there is any takbeer (saying “Allaahu akbar” as in the beginning of prayer) or tasleem (saying salaams, as at the end of prayer) involved in it. The ahaadeeth narrated concerning Sujood al-Tilaawah do not mention anything except sujood (prostration) – “he prostrated and we prostrated with him” – except for one hadeeth which was narrated by Abu Dawood, but its isnaad is subject to question (“he said ‘Allaahu akbar’ when he prostrated”). But there is no tasleem. It was not narrated in any hadeeth, da’eef (weak) or saheeh (sound), that he said tasleem after doing sujood al-tilaawah. If if there is no saheeh hadeeth to indicate that one should say tasleem, then it is not prayer, because prayer has to begin with takbeer and end with tasleem. This is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him). On this basis there is no need for tahaarah (to have wudoo’) or to cover one’s ‘awrah or to face the qiblah. It is permissible to prostrate even if one has broken one’s wudoo’ (minor impurity). Whoever studies the words of Ibn Taymiyah will see that the correct view is that which he expressed, which is that sujood al-tilaawah is not prayer and is not subject to the same conditions as prayer. If you recite Qur’aan from memory and you do not have wudoo’, and you come to an aayah where sujood is required, then according to this view you may prostrate and it is OK. Ibn ‘Umar (may Allaah be pleased with him), even though he was known to be very strict, used to prostrate when he did not have wudoo’, but to be on the safe side it is better not to prostrate without having wudoo’.
Al-Sharh al-Mumti’ by Ibn ‘Uthaymeen, vol. 4, p. 126
On this basis, it is permissible to read Qur’aan after praying Fajr, indeed it is mustahabb. With regard to the issue of the reader coming across an aayah where sujood al-tilaawah is required, after Fajr or after ‘Asr, Shaykh Ibn Baaz (may Allaah have mercy on him) was asked whether the person who is reading Qur’aan after ‘Asr or after Fajr should do sujood al-tilaawah. He said:
It is prescribed to do sujood al-tilaawah after Fajr and after ‘Asr, because this is something that is done for a reason, and because it does not come under the rulings of prayer according to the more correct of the two scholarly views. Rather it comes under the ruling of reading Qur’aan and saying tasbeeh (“subhaan Allaah”) and tahleel (“laa ilaaha ill-Allaah”), so it is permissible for the one who is reading Qur’aan to do sujood al-tilaawah, even if he does not have wudoo’, according to the more correct of the two scholarly opinions.
Fataawa al-Shaykh Ibn Baaz (may Allaah have mercy on him), vol. 2, p. 344