[oleaug]

I am wonndering if I reset my Note 3 back to factory setings, will that break my root or break my phone?
Thanks

[SpookDroid]

I believe it does 'break' root, but not likely to break your phone. It's just like when you receive an OS update. If you install it, you have to root again (if it's an Official ROM, anyway).

[Rukbat]

Factory reset doesn't do exactly what the name implies, it doesn't reset the phone to factory condition, it removes any additions you've made to the phone (apps [which includes rooting apps], contacts, etc.)

SupeSU may - depending on the version of Android and the make and model of the phone - survive a factory reset if you put it in survival mode. But if the phone is unrooted after you reset it and you want it rooted, just root it again. (If you're trying to unroot it, do it manually - remove su and busybox, then uninstall the superuser app, SuperSU, Superuser, whichever one your rooting method installed.)

It definitely won't break the phone unless something goes drastically wrong with the reset function - it's pretty difficult to write software that breaks hardware. (Not impossible, but not easy either.)