There’s always a possibility of rodents, birds, and / or insects penetrating the exhaust system of a furnace. Sometimes is the environment which you may not have much control over, sometimes it could be improper installation of the exhaust system, sometimes it is both.
Some furnace manufacturers recommend protective screens installation on the termination end of exhaust pipes (or even provide them), some don’t. The major problem with a rodent screen installation is that it can cause blockages of the exhaust pipe installed in cold climates where ambient temperature drops below freezing point.
High efficiency furnace exhaust gases contain significant amounts of moisture which will condensate on any type of obstruction inside the vent pipe or at its end. For as long as that exposed section of the vent pipe is in warmer environment everything will be fine but if the temperature drops below freezing point some of the condensate will flow back towards the furnace (assuming the vent pipe has been correctly installed / pitched) and some might start freezing and building up on the grid of the grate / rodent screen, eventually blocking it completely and shutting down the furnace.
I’ve seen some recommendation that protective screens (if not provided by the manufacturer) should be at least 1/2" squares and the squares should be positioned on an angel so they look like an 'X' not “#”. This is probably to improve condensate drainage from the screen.