Well yes, breaking a vein, that is the blue line under the skin, cause serious damage which might be serious like brain disorders. However, normally that is not experienced in most of the cases. What you must be afraid of is dental and gum injury from the piercing itself, or even from prolonged use of the ring or pin.
Just make sure you don't do your lip yourself. If you do the get some help and make sure you do it right. I have mine professionally done. You will indeed break a vein in your lip no matter how careful you are.
Omfg I did mine by myself in school with a tac I got NO infection what so ever and it didnt hurt AT all .
It sounds like you're piercing yourself, do not do this! You can possibly hope to replicate the hygiene and skill levels of a piercing studio in your own home. You'll either pierce it crooked or get a serious infection that will require a visit to the doctor. Have it done by a professional or don't have it at all.