All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, it should be noted that renting should be determined by a given period and if no period is defined for it, like if the two contracting parties agree that the tenant has to pay the landlord such and such [a given amount of money] for each month or each year he used the dwelling unit, then this kind of renting is called Mushaaharah, which is a kind of contract which is not binding to any of the two parties, and if any one of them wants to invalidate the contract he may do so. However, if the renting is eternal, then this is a void renting in principle because among the conditions for the validity of the rent is that the period should be determined. Therefore, according to the two probabilities, it is permissible for your sister to invalidate the contract and drive them out of the house.
However, it is not permissible for your husband to cut relations with the husband of your sister because of this matter, because it is forbidden for Muslims to desert each other without a sound religious reason, as we clarified in Fataawa 86090 and 84976.
With regard to your mother saying about your husband improper words, then she is wrong by doing so, because the in-laws should have good relations and mutual respect. Also, if it is confirmed that your husband mistreats you because of the behavior of your mother, then he is wrong, because it is an injustice to attribute to someone something which he/she did not do, and Allaah Says (what means): {And no bearer of burdens will bear the burden of another.}[Quran 17:15] Therefore, he should fear Allaah and treat you in a good manner and stop harming you. He should know that Allaah is able to take revenge of him, Allaah Says (what means): {But if they [i.e. the wives] obey you [once more], seek no means against them. Indeed, Allaah is ever Exalted and Grand.}[Quran 4:34]
Therefore, we admonish you to advise your husband in a good and soft manner about such a behavior. For more benefit, please refer to Fataawa 86618 and 88304.
As a wife, it is your right that your husband provides you with a separate accommodation where you may live without any problem or embarrassment. He is obliged to provide you with such accommodation even by renting it.
Moreover, it is not permissible for your mother in-law to oblige your husband to live with her, and if she and her husband need care, then it is their children who are obliged to take care of them. In case your husband wants to serve her, he is permitted to do so but you are not religiously obliged to live with her. Rather, you have the right for a separate accommodation as we previously explained in Fataawa 84608 and 86388. In case you renounce this right and live with the parents of your husband as a way of being kind to your husband, then this is good and you may be rewarded for it and it might be a reason for strengthening the marital relations between you and your husband.
Allaah Knows best.