Right after death, the body's muscles are completely flaccid. The jaw opens/goes slack with the direction of gravity. This is VERY noticeable because even in sleep, there is some muscular tension in the jaw.
When I do post mortem care on a deceased patient, I have to close the jaw and secure it with a rolled towel (they actually have "straps" for this) or two, because we don't want rigor mortis to set in with the jaw wide open. You close and support the jaw so that rigor mortis works to keep the jaw closed. The person then appears to be asleep, and it is easier on the family coming to say last goodbyes. A deceased person with their mouth shut either had it closed on purpose or they died in such a position that gravity pulled the jaw closed.
It would have been impossible to close Caylee's mouth unless the tape was applied immediately after death or a few days after death. I wonder if her little jaw was in anatomical alignment with the skull (closed)? I doubt with the time and environmental exposure this would be evident