Does saying どこの出身 sound any different than saying 出身はどこ? More generally, does adnominally modifying a noun have some other feeling associated with in? That is, in comparison to just using a standard subject-predicate construction. (I'm not sure that that is the right descriptor for the second phrasing.)
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(The following is based on something I originally worked out for myself. There may be other views but it seems to hold.)
As the first answer says, the difference between these two expressions will largely depend on context - the situation, how they are said etc. - and the nuance is probably slight but worth understanding.
The particle の is often used to replace が and therefore I would suggest that, to the extent it exists, the difference between 出身はどこ and どこの出身 is similar to the difference between using は and が but with the some provisos.
As が places emphasis on what comes before it and は places emphasis on what comes after, and you have effectively reversed the word order, you could argue that the emphasis on どこ has not substantially changed, even if we extend these expressions into more "complete" formal questions:
The main proviso is (as pointed out in the response to the previous question: noun + の + adjective ) that replacing が with の softens the impact, and places emphasis on what comes after の.
You could analyse this further but given the two expressions are not that different, I would suggest stopping here and moving to look at the difference between between は and が in more extreme and varied cases. There are several explanations. I find the one by Kuno, cited on this site the most useful.
To pick some examples:
Did I get the idea correctly? If so, then I would argue that the difference in feelings of those two are rather subtle/insignificant. If you use them in speaking, your tone of voice would be more influential than the different construction of these phrases.
That said, this subtle difference does exist --- the latter phrases are more interrogating --- because questions in Japanese, in practice, are indicated almost solely by the intonations (i.e. higher pitch to the end) rather than grammars, so the word/phrase that has such a pitch is emphasized as a core of the question.
P.S. as istrasci mentions, どこ出身？ is a common very casual way, but 出身どこ？ is as common as that.