This is not an easy question to answer. I guess it depends on what is meant by Big Bang Theory. For some, the Theory as a whole incorporates the full naturalistic ideas about how the universe sprang into existence without the aid of any outside force. In that sense, the Big Bang Theory is anti-biblical.
The idea of the Big Bang itself, however, may not be. We have pretty solid evidence that sometime in the past, the universe came into being—scientific and Biblical evidence. The Biblical evidence comes from the first chapter of Genesis. The scientific evidence come from various sources such as Einstein’s Theory of Relativity, the observable expanding universe, and others. In fact, the only people today who try to claim that the universe did not have a beginning are those who are trying to escape the philosophical problem they have from the obvious conclusion of the idea that the universe had a beginning: someone created it.
So, the idea that there was a Big Bang at some point in the past is biblical and almost scientifically certain. If God opened his mouth and spoke out the entire universe, or if he snapped his fingers to create it, or if he just decided and it was there in an instant, I’m betting it was loud.
Here’s a short article written J. Warner Wallace of Stand to Reason called The Big Thing About the Big Bang. In it, he describes why the Big Bang is consistent with the Christian worldview, then he speaks to the negative reactions of some Christians who have a hard time disassociating this idea from the naturalistic philosophies that try to deny the existence of God. Take a couple minutes and read it; I think you’ll enjoy it.