I don't believe that it should, but tbh I can't say for certain. My understanding is that you set the time using a high accuracy time source, reset it a month later, then a few months after that, etc and nanosec updates the crystal's drift compensation value which applies corrections to the time displayed, after accounting for the temperature and battery voltage. So it shouldn't be making too many writes, and as long as it isn't checking and correcting the drift too often (like every second or even every minute) it should be fairly minimal additional power draw. I'd be more concerned about excessive backlight usage, but again, this all just me speculating.

On the topic of battery usage though, SWP has a low energy mode, which only updates the face every minute. You can also adjust backlight intensity.