What is the Ruling on a Woman Covering her Hair while Reciting the Qur'an?

There is nothing indicating that it is obligatory upon her to cover it. Rather, Allah's saying in the Qur'an, “Those who make dhikr (remembrance) of Allah (always, and in prayers) standing, sitting, and lying down on their sides.” is general and indicates that it’s permissible for the male and female Muslim to make dhikr of Allah in whatever state they are. An exception to that is where there has come a specific prohibition, for example, it’s not permissible for a Muslim to make dhikr of Allah while he is in the toilet, attending to his need. As for the woman not covering her face or head, there is no Islamic prohibition for it.