Turkey is mostly Islamic and would not be agreeable to anything that would be remotely be considered ‘Jewish’ or ‘Christian’.
They would probably consider the Arks discovery more supportive of Jewish/Christtian theology than Islamic theology and therefore a threat to their culture.
Actually, Islam regards Noah as one of the highest prophets (along with Abraham, Moses, Jesus and Muhammed). I would wager Noah is placed higher in Islam than with Jews or Christians. Obviously we wouldn't place Noah on the same plane as Jesus. And there were many more Old Testament prophets that Jews and Christians revere. Islam basically includes the Pentateuch in their canon and nothing after, except for Jesus.
The Noah's Ark story in Islam is the exact same story as in our Old Testament.