Does one need to be board certified if injecting fillers, Kybella? Or can a doctor's assistant perform these procedures?

The important thing to remember with injectables is that it is a medical procedure so it requires skill and training, but it is also requires an artistic eye. You want to see someone with a lot of experience. Personally, I recommend a board certified plastic surgeon or cosmetic dermatologist. 

Knowledge of anatomy is important when injecting fillers and kybella. I would see a board certified dermatologist or plastic surgeon for these procedures. While other doctors are performing these procedures (radiologists, cardiologists) this is not their field of medicine. You wouldn't want your dermatologist performing heart surgery for example. Doctors do residencies in their respective fields to become experts in their respective fields, this is for a reason. Go to an MD who is an expert in his field. 

YES.  I would always suggest going to a physician for medical procedures such as Kybella.  Knowing the anatomy is critical to avoid complications such as injury to the marginal mandibular branch of the facial nerve.