If the relationship between husband and wife ceases to be sexual in nature due to wife’s sudden loss of interest and lacks love and respect to the extent that the wife and husband do not even sleep in the same room anymore, what, according to Islam, happens to that marriage? Are husbands allowed to seek Mut’a, if divorce is not an option?

First of all, husbands should try to solve the problem, and find out the causes of such failure in relation. They should not forget that wives are human beings, and need the relation also.
However, if the wife is not interested at all, husband can seek mut'a, but should try to get closer to his wife.