If you are using a pose morph it could be that your target morph does not contain the same number if points as the base mesh. Alternatively if it is a weight problem then you need to paint the weights out on the chest presuming that you have used a jaw joint to move the mouth.
From your screen grab it looks like you have used pose morphs…
I have fixed this problem but I thought I might let others know what I did 🙂
I have a character that I have been using for a few years and it's latest incarnation was using R20 and it worked fine. Since I am now using R23, cos I cannot wait to start using the Pose Library, I saved it in R23. Now I wanted to modify the hairstyle which was modeled using geometry NOT hair. I found that I couldn't delete polys or points even when I had deleted the weight tag and the geo was not locked in any way.
If I copied the hair to a new document it worked so I just selected everything in the file and pasted it all into a new file and 'Voila!' I could edit the character's hair as originally intended.
I can only imagine that this must be something to do with the water…
GazzaMataz's post in During face rigging when I'm trying to mouth open chest area is going down ! was marked as the answer