I think you are confused here by the fact that this sentence contains not just an interrogative main clause but also a subordinate "content" clause from which the complementizer that has been dropped. Only the main clause has subject/auxiliary inversion with do-support, not the subordinate clause.
Think about it from the point of view of the answer you expect. In its fullest form that would be something like this
I think [that] it is X's birthday.
Note that there are two clauses here: the main clause is *I think [Y], and Y, the something that you think, is the subordinate clause [that] it is X's birthday.
Now let's walk through the steps you perform to turn this answer into your question.
(1) I think [that] it is X's birthday.
(2) You think [that] it is whose birthday.
(3) Whose birthday you think [that] it is
(4) Whose birthday you do think [that] it is
(5) Whose birthday do you think [that] it is?
- is the answer
- Replace the I with you, and replace the word which you are asking about, X's, with the appropriate interrogative term, whose:
- Move the interrogative term to the front; since it is not strictly a pronoun but a possessive modifying birthday it remains attached to the word it modifies and drags it along with it to the front.
- Since this is a question, you must invert the subject of the main clause (you) with the auxiliary verb. But there's no auxiliary in the original, so you add do to play this role. This is called do-support.
- Now you invert the auxiliary verb do with the subject; add a question mark and you're done.
The interrogative main clause now has the correct inversion, with do support. But the subordinate clause is not affected by this operation—it retains its original form, except that it has "lost" the element which you are asking about.