If you mean formally celebrating the likes of the birthday of Prophet Muhammad (peace be upon him) or Prophet Jesus (peace be upon him) as a devotional act, then this is clearly unlawful. Such festive days are newly contrived innovations that conflict with the dictates of Islamic Law.
Such rites are from the traditions of the People of the Book. They have a devotional purpose and are carried out seeking nearness to Allah. This is the reason I view such occasions to be unlawful and prohibited.
If, on the other hand, you mean celebrating personal birthdays, then this is something different, since it is not intended as a devotional act or an act of worship. Therefore, it is not as serious a matter.
Still, personally I tend to regard it as something disliked for people who are not already accustomed to celebrating this event in their culture, but do so merely to blindly ape cultural practices that are outside of their own experience and understanding.
All the same, birthday parties are not religious occasions and do not have an overtly religious cast to them.
A birthday party is not a festival of the nature of an `îd that is a general festive day for the entire community. It is merely a personal occasion that means nothing more than a remembrance of something dear. Therefore, it is permissible.
And Allah knows best.