The code requires natural draft water heater vent terminations (I’m assuming this is a draft hood equipped WH) to be equipped with a listed (industry approved) rain cap. Also, most rain caps are build in such way that prevents debris or birds from falling into the vent pipe.
If the rain cap (older designs) has no screen or its design does not protect from the vent pipe’s contamination, protective metal screen can be installed (check with local code enforcement division for screen size). Without the cap and screen it can get clogged, most likely by a bird or larger rodent if one can get there somehow. Birds sometimes fall into some unprotected chimneys during the cold season.
What will happen with water that penetrates water heater’s vent pipe not equipped with a rain cap depends on the vent pipe’s shape (offsets or straight run), water heater’s state at the time of rain (burner running or not). The water might partially or completely evaporate on the interior walls of the WH vent pipe, especially it the WH’s burner is running.
It may also run to the base of the water heater, onto the burner, sometimes leak through the vent pipe joints, etc. If the vent pipe connects to the larger, unprotected at the top chimney, rain water often ends up at the chimney’s base, leaking.