YAKKER82 - hi, you can cut out that section of drywall but it'll be a bit of a job. First - is the landlord/owner willing to let you do that? If so, is he/she going to make you hire a pro to come in and do it, then repair the wall?
That section of drywall will be fairly heavy...you'll need to determine how you'll frame the edges to give it support, or else it'll start crumbling/breaking away in chunks and you'll see it collapse in a few minutes of rough handling/transport.
Since there's going to be wall studs somewhere behind your section (normally, 16 or 24 inches apart), you'll need a studfinder to locate them. As electrical wiring may be run behind the wall area you plan to cut out (look for wall outlets near the section you want removed), you may want to buy a studfinder that also senses electrical wiring. They're not too expensive at the big box improvement stores. You'll probably encounter some nails/screws, used to secure the drywall to the wall studs, that will pull out small sections of the drywall and leave holes in it...or as you discover where they're at, you'll have to find a way to pull them from the front (mural side) or to cut them free from the studs. You'll want to use a drywall saw/knife to carefully cut the area of drywall out as you can't simply saw through the wall studs.
Finally, you'll need to get a sheet of drywall the same thickness as that you removed. It'll have to be cut to nearly the same dimensions as the piece you cut out, but slightly smaller. You'll need to "mud" (drywall joint compound and drywall tape) the joints where the new drywall and the old drywall connects. It may take a couple layers of mudding to get it smooth. Then, you'd have to paint the new drywall to match the old drywall, or simply repaint the entire wall.
I've simplified this as I've only repaired smaller holes, no more than 24 inches square. That's why you may end up having to pay a pro to come in and do the drywall work, if the landlord is willing to go along with it. Otherwise, you may end up having to take a few good photos and re-do the mural on something that's more portable, that you can take with you from apartment to apartment. Good luck! Jim D/ West Point, VA