All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Firstly, it should be noted that the present-day pulpit in the Prophet's Masjid is not the one upon which the Prophet, sallallaahu ‘alayhi wa sallam, delivered his sermons during his lifetime. The original pulpit was destroyed in 654 AH when the Masjid was burned, as scholars mentioned. The Haafith Ibn Hajar said: "The pulpit remained as three steps until Marwaan added six additional steps during the caliphate of Mu‘aawiyah …. Ibn An-Najjaar and others said that it remained as such with mild repairs until the Prophet's Masjid was burned in 654 AH; so the pulpit was burned as well." [Fat-h Al-Baari]
As for the Islamic ruling on seeking blessings (Tabarruk) through the pulpit of the Prophet, sallallaahu ‘alayhi wa sallam, before it was burned, some scholars deemed it lawful to touch it for blessing while others deemed it disliked. Ibn Taymiyyah said:
"Ahmad held that it is permissible to touch the place where he (the Prophet, sallallaahu ‘alayhi wa sallam) sat on the pulpit for the purpose of seeking blessing, following the example of Ibn ‘Umar. There are two opinions reported on the authority of Ahmad regarding touching the pulpit for blessings, the most famous one being that it is disliked; and this is also the opinion of the majority of scholars. Maalik and other scholars held that it is disliked to do such things, even if Ibn ‘Umar did it. This is because the senior Companions like Abu Bakr, ‘Umar, ‘Uthmaan, and others never did that. It was authentically reported that ‘Umar ibn Al-Khattaab was traveling when he saw his companions going to a certain place to perform prayer successively. He inquired about it and people said, "It is a place where the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, had prayed." He said, "Are you trying to render the tracks of your Prophet places of worship; those who were before you were destroyed because of that. If prayer becomes due when someone is in that place, then let him pray there; otherwise, let him leave."
Ibn Taymiyyah also said: "Scholars held different opinions regarding the permissibility of placing one's hand on the pulpit of the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, when it was there; Maalik and others disliked it because it is a religious innovation. It has been reported that when Maalik saw ‘Ataa’ doing that, he refused to learn at his hands …. On the other hand, Ahmad and other scholars permitted it because Ibn ‘Umar did so."
For more benefit, please refer to Fatwa 91374 about the permissible and forbidden ways of seeking blessings (Tabarruk).
Allaah Knows best.