Do not put any window type screen inside the vent pipe
– it is not designed for this purpose and it will significantly cut down the chimney drafting capacity. Over time, small screen can become clogged with burning process deposits and dust – it’s dangerous. Instead, install a listed / approved for this purpose screen on top of the chimney.
There are various types of those chimney screens, some are integrated into the chimney cap, others can be added around the existing cap. Just search for “chimney cap”, or “chimney screen” in your browser. If you have a tall, unsupported, hard to access chimney, do not lean the ladder against it!