When humans hear running water and need to go to the restroom at the same time, that isn’t a very good combonation if you know what I mean. When dogs or other animals hear running water, do they need, or want to take care of their business more urgently like humans. Not saying they’re dumb, but are they not smart enough to know the difference or care?
IANA zoologist, but I know that the urination reflex is different in humans and other animals. IOW, humans only urinate to empty their bladders. Dogs, cats, and a host of other animals urinate to mark their territory. Therefore, while I’m sure that a dog can tell if his bladder is full, it’s unlikely that the sound of running water reminds him that it’s time to pee.
Anecdotal Report
Rastakitty’s usual response to running water (say, Mrs. Rastahomie is doing the dishes, or running bath water, etc.) is to swipe at it with his paw. I’ve never noticed that he makes a beeline for the litter box whenever the water is running.