Prohibition on shaking hands with paternal uncle’s wife even if she is old

It is haraam for a man to shake hands with a non-mahram woman
even if she is old. That is because of the general meaning of the evidence
which states that it is forbidden.  

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him)
said: 

It is not permissible to
shake hands with non-mahram women at all, whether they are young or old, and
whether the man who is shaking hands is a young man or an old man, because
of the danger of fitnah (temptation) that it poses to both parties. It was
narrated in a saheeh hadeeth that the Messenger of Allaah (peace and
blessings of Allaah be upon him) said: “I do not shake hands with women.”
And ‘Aa’ishah (may Allaah be pleased with her) said: “The hand of the
Messenger of Allaah (peace and blessings of Allaah be upon him) never
touched the hand of any woman, and he used to accept their oath of
allegiance by words only.” 

It makes no difference
whether one shakes hands with a barrier in between or with no barrier,
because of the general meaning of the evidence, and so as to block the ways
that may lead to fitnah. And Allaah is the Source of strength. 

Fataawa Islamiyyah, 3/76 

What some people believe,
that the wife of a paternal or maternal uncle is not a stranger (non-mahram)
is not correct, and there is no basis for that. Rather the wife of an uncle
is a non-mahram. The question of a man shaking hands with a non-mahram woman
has already been dealt with on this site; please see question no.
2459 and
21183 

With regard to what the
questioner says that not shaking hands will cause resentment and that this
is customary among them, he should refrain from shaking hands with her, and
should explain to her that he is not refraining from shaking hands because
he despises her or thinks little of her, but rather because he is obeying
Allaah and His Messenger. The Muslim has to obey the command of his Lord,
and avoid that which He has forbidden, even if that goes against the
people’s customs. But he should explain to the people that he is doing that
out of obedience to Allaah, and he should be gentle and polite so that he
can convince them to follow the sharee’ah and so that they will not think
badly of him.  

And Allaah knows best.