I've changed the product name in the building settings of my app. It is an existing app, that I want to submit an update for soon. Will this affect how I submit the app?
There is no "product name" in the build settings, so it's a little hard to know what you've changed. Did you change the name of the project?
In any case, the name of the product is merely the name of the .ipa file produced when you do a build. Apple doesn't know or care what that is. My product name often has nothing to do with the names that the public sees, which are:
The name at the App Store (this is not a part of the build process)
The display name in the springboard (this is an Info.plist setting)
For example, take my most recent app, "Diabelli's Theme". The product name, based on the original project name, is (for historical reasons) "MomApp2.app". When I make an archive build, for upload to the store, the archive's name is, for example, "Diabelli's Theme 4-23-13 10.09 AM", and inside it is an .ipa file called "MomApp2". Neither Apple nor the public knows or cares anything about any of that.