Are you :
A piano?
I have keys, but I have no locks. :
piano keys
Some consider me majestic
piano can be majestic
Other find me annoying.
When badly played it sounds injuring
I can do back but not forward :
It is wordplay on Johann Sebastian Bach, the composer and pianist. Forward is not a composer!
I am happy.
I am sad.
I can hold you in suspense.
music transfers emotions, and piano is often used in high suspense movie music (e.g., Psycho's score for instance)
2nd Shot for those three lines (thanks to comments from Mark A, and Peter)
Happy stands for Major Scale, Sad for minor scale, and the suspense bite stands for Suspended scale.