Hi there KassyMiller. I dont keep Whites tree frogs but I suggest separating it immediately from the other frog. The opening and closing of the mouth does sound like it may have been shedding its skin. It would help to post photo's of your enclosure and the sick frog. When did your frog last eat? What do you keep the temp/humidity at?
If you are able a vet would be your best source of professional immediate help as we can really only just speculate. Good luck.