There are only two examples of celebrating a birthday in the Bible. Both are of pagan kings. In Genesis 40:20 we read: “On the third day, which was Pharaoh’s birthday, he made a feast for all his servants“. In Matthew 14:6 it says: “But when Herod’s birthday came, the daughter of Herodias danced before the company.”
However, there is nothing in these stories to suggest that it was wrong for these men to celebrate their birthdays. That a bad man (though it is not clear from the story whether the Pharaoh is bad) does something, is a poor reason for concluding that God forbids it.
That a bad man does something, is not a good enough reason to forbid it. In the same stories, these pagan kings did the following things: get angry; put people in prison; grant mercy; put someone to death; and make a promise. As with actions of all other people, we will have to look at God’s Word to know whether their actions are right or not.
On the subject of birthdays it is very simple: the Bible does not say anything about it. So Christians have complete freedom to celebrate or not celebrate their birthdays. This is determined by culture, not by faith.
It is good to celebrate. Jesus made sure there was enough wine at a wedding (John 2). He told parables about parties (e.g. Luke 14:16-24). He told how to behave at parties (Luke 14:7-11) and how to organize one (Luke 14:12-14). In many cultures, a birthday is as good a reason to throw a party as anything. There is nothing wrong in that, as long as we keep Jesus’ advise from Luke 14 in mind. He basically calls us to not think about our own fun, but to bless others as much as possible. Even in giving and attending parties we should honor God and love our neighbors.