Turning on the Qur’anic recitation while doing something else is better than not turning it on at all. Getting preoccupied with something else in such case is not sinful, for it is not meant to keep one away from the Qur’an. Moreover, paying heed to the recitation is only obligatory while one is performing Prayer or listening to the Friday Sermon; otherwise it is merely recommendable, not obligatory.
Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states: Although we should treat the Qur’an, the Mighty Revelation of Allah, with utmost respect by giving full attention to it while reciting or listening to it, there is nothing wrong for us in listening to it while doing chores that do not require full attention, for the rule of the Shari`ah is that if we cannot have something in its entirety by fulfilling all of its requirements, it does not mean that we should leave it altogether. So even though you are not able to pay full attention to the Qur’an, still you should not be depriving yourself of whatever benefits you can possibly obtain from reading or listening to it in your given circumstance or situation in life.

Moreover, it is definitely far better than listening to music or other mundane activities. Nay, by consistently listening to the Qur’an in this way you certainly stand to gain tremendous spiritual and moral benefits. The same rule applies to listening to the Qur’an or while doing chores at work in the kitchen, etc. In sha’ Allah, as many people would vouch for, such an act, when consistently followed through, is bound to go a long way in elevating ourselves spiritually. Moreover, in this way we will be transforming our physical exercise into a full exercise of body, mind, and spirit.
Let me further add that listening to the Qur’an is an excellent idea to follow by those who are in the habit of jogging or walking as well. Thus they can use the time beneficially by listening to the Qur’an or engaging in dhikr (remembrance of Allah). May Allah make us all among those who fill our waking hours with dhikr as much as possible so that we may be included at least among those about whom Allah the Exalted will say, take out from Hellfire every single person who has celebrated My remembrance at least a single day of their lives. Ameen.