I think you are a:

>! tooth!

I live in a palace where it's dark inside  
No light in there, unless open wide  

>! Teeth live in mouths, which fit this description

Even though I am not a king, sometimes I wear a crown  
I am not a queen either, neither I am a clown  

>! Teeth are neither kings, queens, nor clowns, but they do sometimes have [crowns][1]

I'm known for tearing stuff by moving up and down

>! This describes chewing food

Don't take care of me and I might hurt you

>! If you don't brush and floss your teeth, they'll start to hurt!

Take care of me and get better pictures of you

>! People like to have a dazzling white smile for photos!


>! Without teeth, it would be harder to eat, which would definitely make life hard!

  [1]: https://www.aspendental.com/oral-health/oral-health-and-overall-health