Its not as cut and dry as "he can cry and bleed, he can have DNA", because its an endless argument. Theres nothing in the game that would support or deny the concept of Tidus having DNA.

My guess would be he does.. just because he was dreamed up doesn't mean everything isnt right in that way. I'd think that while he was in existance on spira as a normal person, he was biologically a normal person. But this is definately not one of those situations in final fantasy that can be equated to real life. If anything, the baby could very well either fade out of existance just like its father, or "partially fade" ending in death (can you exist as half a person?)