Can Lip Injections Cause Paralysis?

The type of facial paralysis you are describing is one that likely involves impaired lower lip movement.  This is likely from local anesthetic getting near the nerve that is responsible for that movement and is likely temporary.  A skilled injector who is a specialist in the facial anatomy will not likely have this result.

In my extensive years of injecting experience and being a platinum sponsored surgeon from Allergan, which means I use a lot of injecting materials I have never seen paralysis in a patient. I did not read the story you read but I am guessing the patient may have had a local block in which the injector got local close to a nerve causing temporary paralysis. It is important to go to a skilled injector. While I have several nurses who work in my private surgical center I prefer to inject each patient myself. Being a surgeon allows you years of experience and knowledge of the facial anatomy as well as what you feel will be the most aesthetically pleasing for your patients overall features. Best regards!