There are lots of rumors out there about possible hugely bad side effects of piercings. For example, yes, it is true that an infection on your tongue piercing could possibly travel down your throat and into your heart, to kill you. However, that is like saying any infection on your body can travel like that. It is possible in the same way martians are possible, but not as likely.
This question falls into the same category. When one has a piercing, they go to a professional piercer who understands anatomy. They know what blood vessels, nerves, etc. will be a problem, and watch out for them. I have never heard of any important nerves in the lip region that would paralyze your entire face if hit depending on personal anatomy, but this falls into the realm of possibility. Most likely, someone was pierced, felt numb, and the rumors were passed, growing into paralysis. This is not even a rare problem- as far as I know, it does not exist.