Updated: Jul 28
I painted white over a brown, faux-wood door several years ago. As can be seen in the pictures below, the paint started to chip and even peel along one edge. Also, the wallpaper previously put up was starting to wear off at the wall's corner. To fix both issues, I decided to reorder the wallpaper I had used, in order to fix any patchy areas and go right over this door too.
This is in a high traffic area, likely to get dirty fast so I thought the wallpaper was a better solution to another coat of paint. I got it from Amazon in case you're interested: https://amzn.to/2X3GKpR
It took about 2 full strips to fix the door and 2 corner patches.
I think the end result is great.
This is something not typically done but I'm convinced that a wallpapered door works. Next is some touch up paint around the trim and this area is all set.
I hope this gives you some inspiration for something you can try.