The Prophet (blessings and peace of Allah be upon him) did not deliver the Eid khutbah to the people standing on the minbar, according to the more correct opinion.
Al-Bukhaari (may Allah have mercy on him) said in his Saheeh (2/17):
Chapter on going out to the prayer place without a minbar.
Then he narrated (956) that Abu Sa‘eed al-Khudri said: The Messenger of Allah (peace and blessings of Allah be upon him) used to come out on the day of al-Fitr and al-Adha to the prayer place, and the first thing he would do was to offer the prayer. Then when he finished the prayer, he would stand facing the people, whilst the people were still sitting in their rows, and he would exhort them, and advise them and instruct them. If he wanted to send out a military detachment, he would do so, or if he wanted to issue some instructions, he would do so, then he would leave. Abu Sa‘eed said: The people continued to do that until I went out with Marwan – when he was the governor of Madinah – for Eid prayer on either Eid al-Fitr or Eid al-Adha, and when we came to the prayer place, I saw a minbar that had been built by Katheer ibn as-Salt, and Marwaan wanted to ascend it before he prayed. I grabbed hold of his garment, and he pulled away from me and ascended it, and he delivered the khutbah before the prayer. I said to him: You have changed (the practice), by Allah. He said: What you know has gone. I said: By Allah, what I know is better than that which I do not know. He said: The people were not remaining seated for us after the prayer, so I put the khutbah before the prayer.
Ibn al-Qayyim (may Allah have mercy on him) said:
There was no minbar to be ascended by the imam, and the minbar of Madinah was not taken out; rather (the imam) used to address them standing on the ground. Jaabir said: I attended prayer on the day of Eid with the Messenger of Allah (blessings and peace of Allah be upon him), and he started with the prayer before the khutbah, with no adhaan and no iqaamah. Then he stood up, leaning on Bilal, and enjoined us to fear Allah and encouraged us to obey Him. He exhorted the people and reminded them. Then he went to the women and exhorted them and reminded them. Agreed upon.
End quote from Zaad al-Ma‘aad (1/429)
Ibn Rajab (may Allah have mercy on him) said:
Most of his khutbahs were delivered on the minbar in the mosque, except his khutbah on the two Eids, on the occasion of Hajj, and so on.
End quote from Fath al-Baari (3/403)
See also the answer to question no. 49020.
Such matters should not provoke dissent, dispute or division among the Muslims, and we should not hasten unduly to say that it is an innovation. If that had merely been an innovation, Abu Sa‘eed (may Allah be pleased with him) would have denounced it as he denounced Marwaan ibn al-Hakam for putting the khutbah before the prayer.
Although the established teaching from the Prophet (blessings and peace of Allah be upon him) is not to take out the minbar, hopefully the matter is broad in scope, especially if there is a need to have a minbar in the prayer place for the khateeb.
Ibn Battaal (may Allah have mercy on him) said in Sharh al-Bukhaari (2/554):
Ashhab said in al-Majmoo‘ah: Taking out the minbar for the Eid prayers is a matter that is broad in scope; it may or may not be taken out according to the imam’s wishes. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
Is it Sunnah for the imam to deliver the khutbah from the minbar at the Eid prayer?
Some scholars think that it is Sunnah, because in the hadeeth of Jaabir (may Allah be pleased with him) it says that the Messenger (blessings and peace of Allah be upon him) addressed the people, then it says that he “came down and went to the women.” They said: “coming down” can only be from a higher place. And this is what people do.
Some of the scholars were of the view that delivering the khutbah without using the minbar is preferable.
The matter is broad in scope, in sha Allah.
End quote from Majmoo‘ Fataawa wa Rasaa’il Ibn ‘Uthaymeen (16/350)
And Allah knows best.