What does the Bible say about if one spouse wants the marriage to stay together and the other insists on a separation and what do you do?
Apparently this was a problem in the first century. In the city of Corinth, many people converted to Christianity but their spouses did not. Sometimes this caused problems in the marriage, and the unbelieving spouse no longer wanted to stay married because of the other's new religion. Paul addresses the problem in 1 Corinthians 7:12-16.
If any brother hath a wife that believeth not, and she be pleased to dwell with him, let him not put her away. And the woman which hath an husband that believeth not, and if he be pleased to dwell with her, let her not leave him. For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy. But if the unbelieving depart, let him depart. A brother or a sister is not under bondage in such cases: but God hath called us to peace. For what knowest thou, O wife, whether thou shalt save thy husband? or how knowest thou, O man, whether thou shalt save thy wife?
We can't force an unbeliever to obey God. Sometimes we have to let them be disobedient because they refuse to follow God. Just this week I heard a phrase that isn't found in scripture but seems to be a good one for this situation. When your spouse loses interest in maintaining the marriage, "make God your spouse." In other words, follow God and stay with him to the best of your ability regardless of your unbelieving spouse.