Will a chin implant change where my bottom lip is in comparison to the top lip? (Photo)

Chin implant will not change position of your lower lip unless very large implant is used. However, your problem is not only a small chin? Consultation with an experienced board certified plastic surgeon, craniofacial/ maxillofacial surgeon would help to evaluate extent of your problem and help you to decide what to do. Good luck.

 Hi, I have performed many facial shaping procedures, including Chin Augmentation with dermal fillers or silastic chin implants, for over 30 years.  From the photo, your chin is extremely weak.  When the chin is weak, this creates an imbalance making the nose appear larger, the mid face top heavy, the lower face looks short, de-emphasizes the lips and allows early formation of a "double chin".  
 Proper placement of a silastic chin implant adds forward projection to the chin thereby creating harmony and balance to the lower face.  Using the same incision, liposuction can be performed to reduce the fat and further shape the neck.  Excess skin, from below the chin, can also be removed through the same incision.  
 I have found that placement of a silastic chin implant, through a small curved incision under the chin (also allows excess skin removal) to be very safe, quick, highly effective and far less invasive than a sliding genioplasty (requires extensive tissue dissection, bone cuts and placement of metal screws and plates to secure the cut segments of bone).  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.  In my opinion, you are a good candidate for chin implant surgery.  A properly placed silastic chin implant will add forward projection to the chin but will have no impact on the lips themselves.
Hope this helps.