All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
First, it is crucial to verify the existence of any surviving relics of the Prophet . The authenticity of such claims has been questioned by many scholars, including Ibn Tulun, Ibn Kathir, and As-Suyuti who noted that many relics were lost during various trials (conflicts) throughout Islamic history. Ibn Taymiyyah even asserted that there were none available, and that many claims surrounding them were fabrications motivated by profit and fame.
However, if the existence of any relic is definitively proven, it should be respected due to its connection to the Prophet . Seeking blessings through such a relic is permissible, given its contact with him. This practice is supported by numerous reports from the Companions and their successors. For example: It is narrated in Sahih Muslim that Asma’ bint Abi Bakr brought out a cloak of Tayalisah (a mantle of royal Persian quality) and said: “This was with ‘Aa’isha until she died. After her death, I took it. The Prophet used to wear it, and we wash it for the sick to seek healing by means of it (i.e. the water in which it was washed would be given to the sick to drink to receive a blessing from it).”
However, obeying the Prophet following his Sunnah, and emulating his example are of paramount importance. Doing so brings great requital and enormous rewards and leads to happiness in this life and the next.
Allah knows best.