Visa required?
 

Flying from Istanbul to Punta Cana via Panama City.
Overnight stay required in Panama City.

What happens? What are the possibilities for someone who doesn't have
a Panama visa?

Besides sleeping on the floor in the airport transit area. :(

Are you sure you need a visa? Panama doesn't require visas from most countries. There are a couple of convenient hotel possibilities for you, including a Crowne Plaza just off the airport grounds.

There is no transit area at PTY, one exits their flight like a domestic US connection, free to go straight to any other gate in the terminal.

You don't say what your country of citizenship is.