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.

Most likely, the facial nerve branch innervating the right lip (a.k.a. marginal mandibular nerve) has been stretched or bruised. Unless the nerve is transected, function usually returns to normal. Given the procedures you had, I think that it is unlikely that the nerve has been cut - your surgeon can give you some additional insight into this.
After 6 or 8 weeks, you may see a return of the muscle function. It will be slow, not overnight. I understand that this is stressful for you! Certainly, you want to stay in close contact with your surgeon.
Best of luck for fast recovery!