Well, I did as I said and tested it out. The results were quite interesting. It was pretty much installed as its shown in the picture below, but I rotated it to fit in the little flue grooves. When the pilot light was on, it seemed to be fine. There was definitely a reduction of the amount of convection going on and heat escaping the flue. However, once the burner turned on, it actually picked up the damper and moved it. So, it needs to be tied down or something.
Here is the flue without anything over it.
Here is the flue with the damper.