You most likely know that you can add birthdays, too. (Just look way down at the bottom of the contact page (in edit mode), and tap on “+ field”. This will give you a whole list of possibilities, one of which is “Birthday”.)
And so assuming that you have birthdays recorded for your contacts, you can let Siri tell you when they are.
Just ask her.
Here is how to let Siri help you remember birthdays:
Push and hold the home button to activate Siri.
Ask: “When is _____’s birthday?”
That’s all you do. Now just wait for Siri’s answer.