I have heard that to say 'Eed Mubaarak on 'Eed day, this is Bida'a and therefore Haraam. Please can you clarify this for me?

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 is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

There is no harm on 'Eed day to congratulate someone by saying 'Eed Mubaarak' and other expressions that are said according to customs, and this is not a religious innovation.  We are not aware of any person of knowledge who says it is an innovation.

Imaam Ibn Taymiyyah was asked: 'Does the congratulation and what people utter on 'Eed day like 'Eed Mubaarak' and other expressions, have a basis in Islam or not?  And if it does, then what exactly should be said?'  Ibn Taymiyyah replied: 'The congratulation on 'Eed day like the saying of one person to another when he meets him after 'Eed prayer "May Allaah accept our good deeds and yours", and "May Allaah grant you the reward of this day", and the like, is reported to be said by a group of companions. Some scholars like Imaam Ahmad and others, said that 'this is permissible.'  However, Ahmad said: 'I do not start congratulating anybody, but if I was congratulated by somebody, I answer his congratulation.  Because it is obligatory to respond to a congratulation.  To begin saying the congratulation is not an obligatory Sunnah neither is it forbidden.  So whoever says it is imitating the companions and whoever does not say it is also imitating them, and Allaah knows best.'

Moreover, Shaykh Ibn 'Uthaymeen was asked: 'What is the ruling about the congratulation on 'Eed day, and does it have a specific wording?'  He replied: 'It is permissible to congratulate someone on 'Eed day, and it does not have a specific wording and can be anything that people are used to saying on this occasion as long as it is not a sin.' 

Allaah knows best.