A popular objection to the proofs of natural theology state that the arguments work just as well to demonstrate the existence of the Flying Spaghetti Monster (FSM) and other fictitious entities. For example, the various cosmological arguments reason that some type of First Cause exists. I suppose the atheist's objection allegedly shows that if the argument can be used to show that something we all know to be non-existent exists, then something must be wrong with the argument itself.
This objection fails on at least two counts, in my opinion. For one thing, even granting that the cosmological argument is consistent with a non-existent entity, it simply doesn't follow that the argument is unsound. What if one bites the bullet and says, "yeah, maybe the First Cause is the FSM, or maybe it's something else"? The existence of a First Cause (and a personal one, at that) still stands unrefuted, barring any dealings with the actual premises of the argument.
More importantly, however, the proofs of God's existence (like the TCA) aim to demonstrate the existence of a necessary, unique, eternal and omnipresent being that is simple in its composition. In other words, we are talking about a being whose existence and essence are identical. The FSM, on the other hand, would presumably have an existence not identical to its essence (e.g. noodles aren't identical to existence itself). This means that the TCA cannot be used surreptitiously or to prove the existence of something absurd or non-existent.
Other arguments of natural theology compound the problem. The kalam cosmological argument (KCA) points to the existence of an immaterial, transcendent Creator. Yet, there is nothing immaterial about "noodly appendages." So, via Leibniz' Law, the FSM cannot be the First Cause of the cosmological arguments.