I've had ethernet cable running near my studio equipment and have never had a problem before, but that's just my experience. I've had all sorts of other crazy interference problems I never though possible, so hopefully others can weigh in as well.
I'd imagine the signal strength running through the ethernet cable isn't terribly high (unlike running say, a power cable through with your snake cable), so I'd imagine you won't run into problems. But, unless others come up with confirmation that it will/will not interfere, you might have to do an out-of-wall test (run the snake on the floor with the mics on, and run an active ethernet cable along side it just to see what happens)