So, I’m in deep in the application process and almost done with some of my top choice’s apps 
However, there are a couple of schools that I started the application for (not Common App) but have decided not to submit an application (no fees have been paid, etc). Do I have to call the admissions office to say that I’m withdrawing my application or can I let it sit and expire over time?
No.
You could save yourself possible repeated efforts on their part of trying to call you/resuscitate your status with an email which would have taken you quicker to compose and send than your post here and subsequent readings of replies…
Let it sit. There are stories of students out here that started school specific apps and got acceptance letters even though they didn’t complete the app. :). Doesn’t hurt to just leave it.