Make sure the selector is put correctly on the inside. The Mp5 has a removable part that controls the selector on the inside of the body AND the selector plate on the gearbox. If the selector switch is wobbly, this could be the problem.
I dont think slinky's answer will work, as if your missing that part, the gun won't have a safety. Just two semi modes. HOWEVER, if that piece was screwed in too tight, it could lock it in permanent safe, as he said.
Take the box out, check the switch to make sure it moves freely, then check for both parts of the selector, one inside the body, one on the Gearbox.
-Exr