To Bite Your Lip is to stop yourself from saying anything about something that's bothering you.
For example: Even though Shirley really hated it when Brad teased her about her cooking, she bit her lip. She didn't want to get in a fight with the whole family there for the holiday. Instead, she satisfied herself with the thought that he would be going home soon.
This is not to be confused with bite your tongue, which is similar, but not exactly the same. I will save it for another week to be less confusing. (Think of this as an advertisement of coming attractions. ;) )