Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- You have to find out the actual cause of your hair loss.
2- You can use light fabric for hijab to avoid harm.
3- Do not use tight caps and do not wrap the headscarf too tight.
4- It is permissible to take off hijab when it poses a serious threat to a woman’s health, which may lead to grave consequences.
Answering your question, Dar Al-Ifta Al-Misriyyah, states:
A Muslim woman is to cover her entire body except for her face and hands. It is permissible for her to wear any attire she wishes as long as she abides by the prescribed conditions for hijab and her garments do not lead to tempting men or arousing unlawful desires.
As for your problem, you have to find out the actual cause of your hair loss. Some of the things that cause hair to fall out are stress, poor nutrition, medications, illness, and hormonal problems. The diagnosis will determine the proper treatment.
However, if you continue to experience hair loss, the legal maxim states, “Do not harm or reciprocate harm”.
This means that you can use light fabric for hijab to avoid harm. Do not use tight caps and do not wrap the headscarf too tight; you may tie it a little loosely to protect your scalp from harm.
Based on the above and in reference to the question, it is not permissible to take off hijab. The only permissible case is when the hijab poses a serious threat to a woman’s health, which may lead to grave consequences.
When the necessity ends and the harm is removed, a woman must wear her hijab once again and abide by the full Islamic attire.
Almighty Allah knows best.
Source: http://www.dar-alifta.org/