Is the child obliged to obey the parents and give up his or her share of the inheritance?

The basic principle is that whatever is in a person’s
possession when he is alive belongs to his heirs after he dies. Allah, may
He be exalted, says (interpretation of the meaning):

“There is a share
for men and a share for women from what is left by parents and those nearest
related, whether, the property be small or large – a legal share”

[an-Nisa’ 4:7]. 

Hence it is not permissible for anyone to take the share of
one of the heirs or to demand that he should give up his share of the
inheritance unless he agrees willingly to do that, because that comes under
the heading of wrongdoing and transgression, and the Prophet (blessings and
peace of Allah be upon him) said: “It is not permissible to take the wealth
of a person except with his willing consent.” Narrated by Ahmad, 20172;
classed as saheeh by al-Albaani in Saheeh al-Jaami‘, no. 2780 

With regard to your wife’s mother’s claim that she owns that
part of your wife’s share, if she does not have any proof of the validity of
her claim, then your wife does not have to give up her rightful share of the
inheritance. Not giving up that share is not regarded as disobedience to
parents at all. 

Saying that the mother is not entitled to that part of the
inheritance does not imply that she should not honour her mother and treat
her kindly; rather your wife should speak kindly to her mother and ask good
and righteous people to intervene in order to solve this problem, in the
hope that Allah may guide the mother to truth and right guidance. 

If your wife willingly gives up that part (of the
inheritance), or some of it, to her mother, seeking reward from Allah, then
this is good and an act of kindness, in sha Allah. But, as stated above, she
is not obliged to do so and she is not sinning if she does not agree to her
mother’s request. 

For more information, please see the answers to questions no.
144234 and
178308

And Allah knows best.