Actually, I would think that a more natural way to phrase this question would simply be - "When is your birthday?" Otherwise, a better way would be to change the "what" to "which" in your first question - "Which day is your birthday?" As for the second option, it would be better to change it to - "What is the date of your birthday?", if you'd like to use the word "date" in your question. Just a comment :)

1. "What day is your birthday on?" Yes, this is good. 2. "What date is your birthday on?" This is fine too. I think it sounds more natural to say, "What date is your birthday?" The answers to these questions could be slightly different. In the first one, the reply could be which day of the week (Monday, Tuesday, etc.) your birthday is on this year. In the second one, the answer would be the month and day of your birthday (December 1st, for example).