What is the ruling on raising one's voice with "tahlil" (لا إله إلا الله - There is no god but Allah) and "takbir" (الله أكبر - Allah is the Greatest) and requesting that from people when accompanying a funeral procession?

This act has no basis in Islamic law. What is prescribed when following funerals is to remember the Afterlife and death, and to supplicate for the deceased for forgiveness and mercy without raising one's voice. Indeed, the Companions, may Allah be pleased with them, disliked raising voices during funerals, remembrance (dhikr), and fighting.