Bismihi ta'ala
It is not a sin for a couple to stay married and their marriage be loveless. Marriage is one thing, and love is another. However, a loveless marriage and a cold relationship is not ideal at all. It does affect both sides, and the household as well. Both husband and wife my strive to do something about their feelings towards each other and find ways of re-igniting their love towards each other. It is not healthy at all for them to stay loveless, and it will create a lot of toxicity between them.
In any case, from a fiqhi perspective, if the wife chooses to stay in the relationship under such circumstances, it is not a sin.
And Allah knows best.