What is the ruling on saying aloud "There is no god but Allah, Allah's Countenance is Ever-Abiding" when following a funeral procession, and saying "Ya Rahman, Ya Rahman" (O Most Merciful, O Most Merciful) during burial? And what is the Sunnah in this regard?

The statement "There is no god but Allah, and none remains but Allah" at the time of burial is an innovation (bid'ah), as is repeating "O Most Merciful" (Ya Rahman). The Sunnah for one who follows a funeral procession is to reflect and ponder on their own ultimate end. What is legislated upon completing the burial is to seek forgiveness for the deceased and to ask for their steadfastness.