Does he have to obey his mother by making his wife go out to work?

Firstly: 

It is permissible for a woman to work outside the home so long as her work is regulated by Islamic guidelines, although it is better for her to stay at home unless she needs to work, because Allaah says (interpretation of the meaning):

“And stay in your houses, and do not display yourselves like that of the times of ignorance, And perform As‑Salaah (Iqaamat‑as-Salaah), and give Zakaah and obey Allaah and His Messenger”

[al-Ahzaab 33:33]. 

Although this is addressed to the wives of the Prophet (blessings and peace of Allaah be upon him), the believing women follow them in that. Rather it is addressed to the wives of the Prophet (blessings and peace of Allaah be upon him) because of their honourable status with the Messenger of Allaah (blessings and peace of Allaah be upon him) and because they are examples for the believing women. Hence the verse contains commands to all women:

“And perform As‑Salaah (Iqaamat‑as-Salaah), and give Zakaah and obey Allaah and His Messenger”

[al-Ahzaab 33:33]. 

This is also indicated by the words of the Prophet (blessings and peace of Allaah be upon him): “Woman is ‘awrah (that which must be concealed), and when she goes out the Shaytaan’s hopes are raised. She is never closer to Allaah than in the inner part of her house.” Narrated by Ibn Hibbaan and Ibn Khuzaymah; classed as saheeh by al-Albaani in al-Silsilah al-Saheehah, 2688. 

And the Prophet (blessings and peace of Allaah be upon him) said with regard to women praying in the mosque: “… but their houses are better for them.” Narrated by Abu Dawood (567) and classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

See the answer to question no. 22397. 

Secondly: 

You are not obliged to obey your mother by making your wife go out to work, because obedience is only required in that which is right and proper, and when women go out to work, in most cases it is not free of some negative consequences for her, or causing the rights of her husband and children to be affected. Moreover, it does not seem that there is any benefit for your mother in that either. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said in al-Ikhtiyaaraat (p. 144): A man is obliged to obey his parents in anything that does not involve sin, even if they are evildoers … this applies to that in which there is benefit for them and no harm for him. Even if it is difficult for him but does not harm him, it is obligatory, otherwise it is not. End quote. 

If one or both parents tell their son to do something in which there is no benefit for them, then he is not obliged to obey them. 

You have to treat your mother kindly and explain to her that you do not need your wife to work, and that her staying home is more proper and more beneficial. 

May Allaah help us all to do that which He loves and is pleased with. 

And Allaah knows best.