Does a husband have to be patient with his wife all the time and not divorce her?

Undoubtedly the wives of Nooh and Loot (peace be upon them) will enter Hell with those who will enter it, but they did not commit any obvious sin that would imply kufr, otherwise it would not have been permissible for these Prophets to have remained married to kaafir women, because Allaah says (interpretation of the meaning):

“… Likewise hold not the disbelieving women as wives” [al-Mumtahanah 60:10] 

Perhaps the wife of Nooh was concealing kufr, or perhaps, despite the fact that Nooh had been calling people to Allaah for so long, she was influenced by the call of her people when she saw that all of her people were following kufr, so she became doubtful and wondered how he alone could be a believer when all of these people were disbelievers, and they formed the majority of their nation. So her kufr may have been secret. The same applies to the wife of Loot, of whom they said that her only sin was that she told her people about his guests, i.e., she called them to come and commit obscene actions with them. This was her sin, but it is possible that she was also a kaafir in secret. Hence Allaah said (interpretation of the meaning):

“… except his wife, she will be of those who remain behind” [al-‘Ankaboot 29:32] 

This is a summary of the response given by Shaykh ‘Abd-Allaah ibn Jibreen, may Allaah preserve him. 

It is permissible for a husband to divorce his wife so long as there is a shar’i reason for doing so, such as a lack of religious commitment, a bad attitude, lack of chastity, negligence, etc., even if she is not a kaafir. But if she is a righteous believer, let him keep her, even if he dislikes some of her characteristics, as the Prophet (peace and blessings of Allaah be upon him) said. It was narrated that Abu Hurayrah said: the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Let not a believing man hate a believing woman. If he dislikes one of her attributes, he will be pleased with another.” (Narrated by Muslim from Abu Hurayrah, 1469). 

When the Prophet (peace and blessings of Allaah be upon him) wanted to divorce Hafsah, Allaah revealed to him: Go back to Hafsah, for she fasts a lot and prays a lot at night, and she will be your wife in Paradise. Al-Mundhiri said: this was narrated by al-Nasaa’i and Ibn Maajah. ‘Awn al-Ma’bood Sharh Sunan Abi Dawood, hadeeth no. 2283. 

The husband has to strive to reform his wife and pray to Allaah to reform her. Allaah will reform a wife in whom there is some crookedness, if He wills, as He said concerning His slave Zakariya (interpretation of the meaning):

“… and [We] cured his wife for him…” [al-Anbiya’ 21:90]

 Some of the mufassireen (commentators) said that she used to have a sharp tongue, i.e., her speech towards her husband was offensive, so Allaah reformed her.

 A man may put up with the difficulty of keeping his wife in order to ward off a greater difficulty, which is that of separating the children and dividing the family. But if the harm caused by staying with one’s wife is greater than the harm caused by separating from her, there is nothing wrong with him divorcing her. And Allaah is the Source of strength.