The major flaw in this objection is that it assumes what it claims to prove. The power lies in suggesting that a particular belief is irrational because it is based on a prior belief in Christianity rather than the truth. However, to make the objection is to assume that Christianity is itself irrational and false. If Christianity were true, then the objection would have no teeth. Imagine that someone said, "you just believe that the earth rotates around the sun because you live in the 21st century." That wouldn't phase you for a moment. Why? Because it is true that the earth rotates around the sun. You are convinced by the evidence that it is a matter of fact. It doesn't really matter why you believe it. And there are very likely some people living in the 21st century who, for whatever reason, still believe that the sun rotates around the earth. So the fact that we live in the 21st century is really not relevant to the question of whether or not it is true. It only tells us how or why we believe it, but not whether or not it is true. Suppose that I said to one of my atheist friends, "You only believe that because you're an atheist." I am certain that they would immediately snap back and say something like, "No, I believe it because it's true, and here are the reasons..." And that is an appropriate response. The important question isn't why they believe it, but whether or not it is true.
In the end, the objection, "you only believe that because you're a Christian," boils down to "Christianity is false and irrational and so I don't have to take you seriously." It simply assumes that the belief in question is false because of a prior belief that Christianity is false. Both sides are working from a prior view of the truth value of Christianity. You could almost turn it back around and say, "You only don't believe it because you're not a Christian." But the important question isn't why you believe it, but whether or not it is true. So that is how I think we should respond to it--simplify the discussion to the question "is it true?" I already mentioned one way to do that: by saying "No, I believe it because it's true, and here are the reasons..." Or an even better way, in my opinion, is to ask them for the reasons why they think it is false. You aren't the only one making a claim, so you aren't the only one who bears the burden of proof. They may disagree with your view on the issue, so you ought to be able to give reasons for your position, but by raising this particular objection, they are claiming that your position is unquestionably false--not even worth consideration. They need to give reasons to back that up.