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

Is it permissible to shake hands with a paternal uncle’s wife if she is old, and if not shaking hands will provoke rancour and resentment, because that is the custom among relatives?.
Praise be to Allah.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 Baz (may Allah 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 Allah (peace and blessings of Allah be upon him) said: “I do not shake hands with women.” And ‘Aa’ishah (may Allah be pleased with her) said: “The hand of the Messenger of Allah (peace and blessings of Allah 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 Allah is the Source of strength. 
Fatawa 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 Allah 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 Allah, 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 Allah knows best.