Can I remove the text beneath apps on the iOS 7 home screen (SpringBoard)? The text beneath the apps looks really out of place with the new iOS 7. Is there any way to turn it off without jailbreaking?
No this is not possible. Not in the standard iOS package anyway. 

I have no experience with jailbroken iOS, but I would assume so, although there is no JB for iOS 7 just yet.