Nothing happens to us if we do not have free will. If you want to argue that we have no free will, it is necessary to say that we never have had free will. Therefore, our lives up to this point have been lived without us having any free will. If that is the case, then nothing will change.
That being said, it would change things if we came to believe we have no free will. If people think that their fate is already determined, they might act in hedonistic and anarchic ways. They would simply do whatever they want, saying their actions have been predetermined. Therefore, if we start believing we have no free will, we might behave worse than we do now.
We'd live just as we are living now, everything pre-determined, even our own thoughts, decisions, and choices.