What is the ruling on one who tells his wife, "If you go to your mother and greet her, then you are divorced," or prevents the wife's relatives from entering to see her? And must the wife obey him in this situation?

It is disliked for a husband to make his wife's divorce contingent upon her visiting her mother and greeting her, due to the risk of severing the marital relationship. This is because, according to the majority of scholars, a conditional divorce takes effect if the condition is met, even if the husband did not intend divorce. Sheikh Al-Islam Ibn Taymiyyah, however, holds that an expiation for breaking an oath (kaffarat yamin) is required if the husband did not intend divorce.

As for the husband preventing his wife's relatives from visiting her, there is a detailed explanation:

Hanafi and Maliki schools: The husband may not prevent the wife's parents from entering to visit her every Friday, nor may he prevent other mahrams from visiting her every year, nor may he prevent her young children from another (marriage) from visiting her every day. If he accuses her parents of corrupting her, then a ruling is issued allowing them entry with a trustworthy woman from the husband's side. Shafi'i school and a view within the Hanafi school: The husband has the right to prevent entry into his home because it is his property. Maliki school: A ruling is issued allowing her parents and her adult children from another (marriage) to visit her in the marital home once every Friday. Hanbali school: The husband does not have the right to prevent her father from visiting her, as this would constitute severing kinship ties. However, if there are circumstantial indications that their visit would cause harm, then he has the right to prevent it.

The wife must obey her husband if he prevents her relatives from visiting in his home, provided he has the right to do so, due to the general prohibition against allowing anyone into the marital home whom the husband dislikes.