The question would be did Cooper use the emergency system? if he did a hole in the wall would of been noticed. if he didn't how would the FBI or anyone else know a placard was missing? if Tina didn't know about the panel or didn't tell him, she would open the door to the normal stairs blocking the emergency panel.
Talking with Eric my assumption the placard was similar to a decal or a bumper sticker were correct. it has adhesive on the back. how did it get off the wall? pressure sure the hell isn't going to do it..