Praise be to Allah.
There is nothing wrong with putting oil on the hair during the day in Ramadaan and that does not affect the fast.
Shaykh Ibn Baaz (15/259) was asked: what is the ruling on using kohl and cosmetics for women during the day in Ramadaan? Does that break the fast or not?
Kohl does not break the fast for women or for men, according to the more correct of the two scholarly opinions, but it is better to use it at night for one who is fasting. The same applies to things that are used to beautify the face such as soap and creams etc, that are applied externally to the skin, such as henna, make-up and the like. There is nothing wrong with the fasting person using them, but make-up should not be used if it is harmful to the face.
Shaykh Ibn ‘Uthaymeen said in Fataawa al-Siyaam (228):
All kinds of creams, whether used on the face, back or elsewhere, have no effect on the fasting person and do not break his fast.
The Standing Committee (10/253) was asked: Does kohl or cream used by a woman during the day in Ramadaan break the fast or not? They replied:
Whoever uses kohl during the day in Ramadaan when fasting does not invalidate his or her fast. The same applies to one who puts oil on the hair during the day in Ramadaan – it does not break the fast.”.