If I delete my backup on my old phone, will anything happen to my new phone?
No it shouldn't, because the backups don't affect anything anyway.


  The short answer is no—deleting your old iPhone backup from iCloud is completely safe and won’t affect any of the data on your actual iPhone. In fact, even deleting the backup of your current iPhone won’t have any impact on what’s actually on your device. The information stored in your iCloud Backups is just that—a backup, or copy, of what is currently on your iPhone.


From http://www.ilounge.com/index.php/articles/comments/deleting-old-icloud-backups