Edit: Thanks for posting the photo, answer updated (see below).
Here is some general advice.
Since larger chunks of wood are missing, use wood filler to fill it in (found at any home improvement store). Use a putty knife to match the shape of the trim as closely as possible. After the filler cures, use sand paper to round it out.
You will also need to paint over the filler. Get a chip of the wood trim and take it with you to the home improvement store and get them to color-match the paint (you may also be able to pull a piece of the trim off the wall).
The other option would be to find similar trim at the store and replace the damaged piece. Some fancy trims can be tough to match, but what you have looks pretty basic.
For what it's worth, in the last place we rented, our rabbit used to gnaw on the door trim in our bedroom. We eventually trained the rabbit not to do that, but it took a while, and the damage got pretty bad. We repaired the trim using the above methods. We got our full security deposit back.
I am going to make the assumption that this condition has developed over time, rather than been this way since built. I am also assuming you meant an interior door, not an exterior door, since it is to a laundry area, not an entry door.
The easy answer would be to trim the door to fit better. But before we go that route, let's try to determine why the door has sagged. Open the door half way, stand in front of the handle edge, grab the knobs on both sides and try to lift the door. Is there any give? If you are able to lift the door, which would effectively close the gap you described in your question, then we have a hinge or frame issue. If you are able to lift the door to a point where the door would close properly and notice any give to the frame, here is a fix. Remove the center screw from the jam side of the top hinge. Use a 3 to 4 inch screw of similar finish to replace the one removed. This screw should reach the jack stud and can be used to move the top of the door frame closer to the jack stud, evening out the gap on the top of the door. Adjust this screw until you get the proper fit.
If the above mentioned technique doesn't work in your case, then trimming the door may be necessary. If so, carefully measure how much needs to be trimmed off, scribe a straight line on both sides of the door. Remove the door, secure it on it's side (just so it is easier to reach and work on,) and use a belt sander, plane, or DA sander to remove material just up to the scribed line. This will give you a nice straight edge. Re-hang your door and enjoy!
Best Answer
I would seek to avoid affecting the embossed grain, which will only make the appearance worse. Gently press the upper side of the split to see if you can get it to level out. If necessary, use a razor knife to trim away the interfering fiber material, allowing the surface to come back into alignment.
Then I'd rub the crack with wood glue to bond it together. Let that dry, then apply a wood filler in the remaining void. Before the filler dries, rub it out with a cloth so that the only remaining material is in the crack itself. Remove all material from the painted surface.
Finally, touch up the paint as minimally as possible. Use a small art brush and only paint the damaged line. Wipe any paint off the undamaged surface.
As a rental property owner I'd be satisfied with such a repair done well. It's expected that minor damage will occur in a residence, and this is about all that can be done short of replacing (or re-skinning) the door.