Praise be to Allah.
Listening to the Holy Qur’an and paying attention to it when it is recited is recommended according to Islamic teachings. There is a difference of scholarly opinion as to whether that is obligatory, and there are two views, the more sound of which is that it is not obligatory, except in the case of prayer, when it is obligatory. This is the view of the majority of scholars.
But the Muslim should be keen to listen attentively when the Qur’an is recited, and not let his attention wander from it, except in the case of work or need. This comes under the heading of venerating and respecting the Qur’an. We have discussed this previously in the answer to question no. 88728.
With regard to leaving the broadcast of the Holy Qur’an playing whilst you are in the house or out of the house, awake or asleep, there is nothing wrong with that, so long as there is nothing happening at the same time that would drown it out, and there is no noise, idle talk or arguments, and so long as it is not in an inappropriate place. In such cases it is more appropriate to turn off the broadcast. This comes under the heading of venerating and respecting the Qur’an, which is something enjoined and encouraged.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: What is the ruling on having a radio or cassette recorder playing the Holy Qur’an in the house, when going out of the house to visit family or relatives?
He replied: There is nothing wrong with that, so long as there is no idle talk going on around it. If there is someone nearby who is engaging in idle talk and speaking and arguing, then that is not right; it is better not to leave the radio on in that case, and it should be turned off, because that is a kind of showing disrespect towards the Qur’an. But if the radio is on when there is no one around, or there is someone who is listening, or keeping quiet, or sleeping, then there is nothing wrong with it. End quote.
And Allah knows best.