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.