Apple ID, same ID on two phones...

Hi, I have a new iphone which is my primary phone, and I use my old iphone for streaming movies. They both have the same Apple ID. Can I delete the Apple ID off my old phone? But keep it on the new one?

One of the problems is when I use Find My Phone, (activating a sound if phone missplaced) the sound is transmitted to my old phone, not my new phone!

Many thanks