How can I cover a wall that's painted in rented apartment?

I would hang fabric. You could even use sheets or painted drop cloths.

Why do you want to cover the wall? Is it a horrible color or finished in a faux finish of some kind? If you can put in a few nail holes, I suppose you could mount a tension rod at the top and hang a full length curtain. Make it as full as possible so it looks like nice drapery. You could also make a casing at the bottom and run a tension cord at the bottom too. Can you give us a bit more info regarding the wall, such as (a) what you dislike about it and (b) what size is the wall you want to hide then maybe we can help more. Thanks.

Use wall paper.

You can use fabric, liquid starch, and put it up like wallpaper. Soak the fabric in liquid starch, wring it out, and "wallpaper" the wall. Smooth out with a sponge. It can easily be peeled off and the starch washed off with no damage to the wall.

I like the drape effect, but you could hang all the same size mirrors ( many, perhaps) the same height that would take away the color of the wall and bring in more light from the reflection.

I'm speaking of the oblong mirrors that you can find at Walmart that people usually use in a dressing room or back of doors. Cheap, but easy, and could be an interesting effect.