Paralyzed Lip After Neck Lipo and Chin Implant

There are two major nerves that can be effected by chin augmentation.
One is a motor nerve that is typically injured/bruised from the liposuction (as mentioned by Dr. Yuan) and typically accompanies chin augmentation. Injury will result in the appearance you demonstrate in the picture. Liposuction will bruise the nerve temporarily with expected recovery, whereas surgery such as a facelift may actually cut the nerve, in which case recovery may not be possible. This does not seem to be the case in your specific instance.
The other is a nerve coming directly out of the jaw bone, which provides feeling to the lower lip. There is one on both sides. Injury to this will cause numbness and tingling in the lower lip but should not effect the movement of it.
You should always keep an open dialogue with your surgeon and express your concerns.
I hope this helps.

This type of nerve paresis (or weakness) can occur after liposuction alone or with a chin implant. It could also be a muscle injury as well. It usually comes back with time but may be a month or longer.

This type of injury will typically correct itself over time - usually within 3 months or so.  Some believe that continuing to try to move the paralyzed side may help speed up recovery.