DIAGNOSIS The door switch—a small plunger or button that gets pressed when the door closes—is either stuck in the 'on' position, misaligned, or has failed. It's usually located at the top of the door frame inside the cabinet.
FIX Unplug the fridge for safety. Open the door and look at the top front edge of the interior cabinet. You'll see a small plastic button or lever. That's the switch. 1) Press it with your finger a bunch of times to see if it's stuck. You might hear a faint click. 2) Look for any debris (like a bottle cap or label) that might be jamming it. Clean around it with a dry Q-tip. 3) Check the alignment. When you close the door, the part of the door that should hit the switch might be missing it. You can sometimes gently bend the switch bracket forward a tiny bit with pliers so the door makes contact.
IF THAT DIDN'T WORK If the switch moves freely and the door hits it but the light stays on, the switch itself is likely broken. This is a simple, inexpensive part. You can find the part number by looking at the switch and contacting Manastin or an appliance parts site. Replacing it involves a couple of wires and a screwdriver.
HEADS UP Having that light on 24/7 generates a little heat inside the cabinet, making the compressor work harder. It'll also burn out the LED bulb faster. It's worth fixing sooner rather than later.