I have an internal fire exit door which doesn't fully close and one final exit door which has been damaged due to having the self closer mechanism removed, as these are fire exits and not fire doors, does this pose a problem?

If it's an internal fire exit from one internal area to another there is a good chance it's meant to be a fire door as well in which case it needs repairing/replacing and a new closer fitting.

The final fire exit to open air doesn't usually need a closer as it's not a fire door unless protecting an adjacent external escape route.