What is the ruling on shaking hands with worshippers after the prayer, given the insistence of the elderly on this custom and the Imam's feeling of embarrassment when refraining from it?

Shaykh al-Islam Ibn Taymiyyah stated that shaking hands after prayers is an innovation (bid'ah), so this should be explained gently to these individuals. Handshaking is a Sunnah when Muslims meet, but there is no meaning in shaking hands with those who are already seated in the same place.