Firstly, since "Geburtstag" is masculine, it would be "dein Geburtstag". Secondly, no, you generally can't say it like that. Even though it would be understood, it's just not something a native speaker would usually say.
Kind of like how saying "When do you have birthday?" might be understood in English, but you'd definitely be recognised as foreign!
Wann hast du Geburtstag is the normal way of saying it. In a few regions you also hear Wann ist dein Geburtstag, but most of the time this is considered a small mistake.
"Wann hast du den dein Geburstag?" Is this correct? If it is literally translated, it can be used this way in Greek.
In English we say "When's your birthday?" often. Duo marked this wrong. Duo is wrong
Duo's completely inconsistent when it comes to contractions. Report it as they do eventually seem to get fixed. It wouldn't accept 'When's it your birthday' or even 'When is it your birthday' either.