How can I repair scratches on door molding?
The smaller scratches could easily be filled with wood filler and painted. 

The only decent way to fix the worst ones is to replace the trim, IMO. Or you could just paint it as is and hope that noone notices.

The trim seems to be made of MDF.I would also argue that this damage could still fall under normal wear and tear. I would never use mdf for trim in my rental units. It just doesn't last.