What happens to your lips when you get braces to fix your overbite?

You ask a very good question.
Here are a few things to consider before you make a decision.
No overbite "needs" correction.  Crooked teeth and overbites are not a disease, they only "need" to be corrected if the patient desires a correction.  On the right person, A little bit of overbite can look a bit sexy, especially with full lips   Open any women's fashion magazine for proof.  That said, if YOU want to show less over bite, here is an answer to your question.
Moving the teeth forward or backward will change the position of the lips, but it is not a 1-1 relationship.   With thicker lips, there has to be more movement of the teeth before you begin to see movement of the lips.  That said, often times the pouty little curl of a lower lip comes having a slight overbite.  The curl comes from part of the lower lip getting slightly trapped behind the upper teeth which causes the front of the lip to roll slightly.  Correct the overbite and the pout will decrease noticabley. 
If you are concerned with this level of detail, you need to get an opinion from an experienced orthodontist.  The average orthodontist starts 200-500 patients every year while dentists who do orthodontics see only a fraction of that.  Orthodontists live and breath lip position and facial balance and you want that kind of experience behind any opinion that influences the decision you make.
After 20 years of doing this, my experience tells me that for me to consider any treatment on you, I would have to hear you say "I hate my teeth more than I love my lips"  before I would even consider changing anything.  Overbite is only a problem if YOU think it is a problem.