Firstly: 

The wife is obliged to obey her husband because he is in charge of her. Allah says (interpretation of the meaning): “Men are the protectors and maintainers of women” [an-Nisa 4:34]. 

Ibn Katheer (may Allah have mercy on him) said: ‘Ali ibn Abi Talhah said, narrating from Ibn ‘Abbas: “Men are the protectors and maintainers of women” means: they are in charge of them, that is, the wife should obey the husband in that in which Allah has commanded her to obey him; part of that obedience is that she should treat his family kindly and look after his wealth. 

Something similar was stated by Muqatil, as-Suddi and ad-Dahhak. End quote from Tafseer Ibn Katheer, 2/293 

Two things are excluded from the obligation of the wife to obey her husband: 

1.     If obeying him will result in falling into sin, either by failing to do something obligatory or by doing something that is forbidden. In this case it is not permissible for the wife to obey her husband, because the Prophet (blessings and peace of Allah be upon him) said: “There is no obedience to any created being if it involves disobedience towards Allah, may He be glorified and exalted.” Narrated by al-Bukhari, 6830; Muslim, 1840

2.     If obeying the husband will result in harm to the wife or it will lead to loss of her rights. In that case she is not obliged to obey her husband, because the Prophet (blessings and peace of Allah be upon him) said: “Obedience is only in that which is right and proper.” And he (blessings and peace of Allah be upon him) said: “There should be neither harm nor reciprocating harm.” 

For more information, please see the answer to question no. 97125

Secondly: 

It is permissible for a woman to cut and dye her hair, so long as that cut does not resemble men or kafir women or immoral women, and the hair is not dyed black. 

The ruling on cutting and dyeing the hair for women has been discussed previously. Please see the answer to question no. 139414 and 82671

If the husband has not given his wife permission to cut or dye her hair, then it is not permissible for her to do that. That is because she is enjoined to obey him so long as he does not tell her to do anything sinful and because it is the husband’s right that his wife should beautify herself for him. There is no doubt that the hair is part of the woman’s beauty. 

And Allah knows best.