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.