There isn't one right answer to this question, but here are some things to keep in mind:
With the right amount of horizontal scaling, it is quite possible you could keep scaling out use of the debug server forever. When exactly you would need to start scaling (or switch to using a "real" web server) would also depend on the environment you are hosting in, the expectations of the users, etc.
The main issue you would probably run into is that the server is single-threaded. This means that it will handle each request one at a time, serially. This means that if you are trying to serve more than one request (including favicons, static items like images, CSS and Javascript files, etc.) the requests will take longer. If any given requests happens to take a long time (say, 20 seconds) then your entire application is unresponsive for that time (20 seconds). This is only the default, of course: you could bump the thread counts (or have requests be handled in other processes), which might alleviate some issues. But once again, it can still be slow under a "high" load. What is considered a "high" load will be dependent on your application and the expectations of a maximum acceptable response time.
Another issue is security: if you are concerned at ALL about security (and not just the security of the data in the application itself, but the security of the box that will be running it as well) then you should not use the development server. It is not ready to withstand any sort of attack.
Finally, the development server could just fail outright. It is not designed to be used as a long-running process (days, weeks, months), and so it has not been well tested to work in this capacity.
So, yes, it has limitations. Yes, you could still conceivably use it in production. And yes, I would still recommend using a "real" web server. If you don't like the idea of needing to install something like Apache or Nginx, you can still go with a solution that is still as easy as "run a python script" by using some of the WSGI Standalone servers, which can run a server that is designed to be in production with something just as simple as running python run_app.py in the command line. You typically just need to create a 4-5 line python script to import and create the server object, point it to your Flask app, and run it.
gunicorn could be run with only the following on the command line, no extra script needed:
gunicorn myproject:app
...where "myproject" is the Python package that contains the app Flask object. Keep in mind that one of developers of gunicorn would probably recommend against this approach. See https://serverfault.com/questions/331256/why-do-i-need-nginx-and-something-like-gunicorn.
