Shaykh Muhammad ibn Saalih al-Uthaymeen rahimahullaah in his Majmoo Fataawaa vol. 16 pge. 244 was asked the following question:
Q: What is the ruling of the extra takbeers (saying ‘Allaahu akbar’) in the Eid prayer? And what is to be said between the takbeers? And what is the ruling on raising the hands with each takbeer?
A: The ruling of the extra takbeers (saying Allaahu akbar) is Sunnah (recommended); if a person was to do that then he would be rewarded – and if he did not do them then there is nothing upon him.
However it is not befitting that a person leaves them off, so (by doing the extra takbeers) the Eid prayer is distinguished from other than it (from the prayers).
As for what is to be said between them: Then the scholars have mentioned that one praises Allaah, and sends blessings upon the Prophet sallAllaahu alaihi wa sallam; but if he were to leave that off then there is no harm.
As for raising the hands with every takbeer (saying Allaahu akbar) – then it is sunnah (recommended).