Officially, neither the birthday is revealed, nor has it really played a role in the plot.
However, Aoyama said in an interview, that there is an "obstacle" preventing him from revealing Ran's birth date, but since the birthday hasn't passed Ran is 16 right now"
The interview, quoted from Detective Conan World's forum:
Tell me about Ran's birthday! → Since there's an obstacle, I can't tell it for now. But since the birthday hasn't passed, Ran is 16 right now.
I think that a future volume will be about her birthday, so everything else would be wild speculation.