You certainly have weakness of your left lower lip, and this is one of the possible problems that can happen after Chin Implant surgery. Any surgeon that performs this operation regularly has seen this problem. It's not common, but the function of your lower lip will return provided the nerve wasn't cut or cauterized. This typically occurs from stretching the nerve while placing the implant.
I've seen this problem (fortunately not frequently), and it tends to normalize in less than 6 weeks. It could take up to 6 months, but that is not the norm.
If it's not better in a month, you can have a small amount of BOTOX injected in the right lower lip (5-10 units). This will provide symmetry if the problem persists for over a month.
I'm sorry you're going through this, but you should feel confident that in all likelihood, your smile will completely go back to normal.