Six of the months require the definite article when used with mí (Mí na Feabhra, Mí an Mhárta, Mí na Bealtaine, Mí an Mheithimh, Mí na Samhna, Mí na Nollag). Feabhra, Bealtaine, Samhain and Nollaig are all feminine, so they take na in the genitive ("the month of X" is a genitive construction).
You can find out more here.
The NEID reflects current usage, and it looks as though some people are using all of the months without the definite article, except for Mí na Samhna and Mí na Nollag.
(The FGB and the NEID both give Mí Feabhra as the only option, and the NEID offers Mí Márta, Mí Bealtaine and Mí Meithimh, which certainly makes life easier!)
It's days of the week, not months, that have Dé in front of them
each day has at least two forms, one for the day as a subject or concept (like An Luan, "Monday", lit. “the” Monday) and one used to say when something is happening (like Dé Luain, lit. “on the day of” Monday).