"To earn Hilton Honors Points or credit toward tier status for hotel stays, a Member must be a registered, paying and staying guest at a participating hotel within the Hilton Portfolio and meet all of the conditions described herein."
Housekeeping would soon notice you weren't a staying guest and as stated in the rules, you do have to be a staying guest. You could probably get away with not staying the first night, but trying for 10 nights is pushing your luck a bit much.
The problem isn't really that you must actually stay. The issue is that, once housekeeping finds an empty room, you might be automatically checked out of the hotel. Leaving a "do not disturb" sign is irrelevant for stopping this.