Would a torn rotator cuff cause chest bone/collar bone to ache?

Depends: The answer depends upon where the tear occurred and it's severity; then there is always the referred pain component from the event...So i'll say yes.

Not usually: Your rotator cuff muscles and tendons reside deep to your clavicle (collar bone). Muscles that attach to your clavicle include the trapezius, a portion of the pectoralis and deltoid muscles to name the more common ones.