Now, you might be asking, "What could possibly be wrong with that?" There are three primary problems that need to be accounted for, and I think they are strong enough reasons to reject natural theology as a valid discipline.
First: Nature (Creation) itself is fallen. While it is true that Nature does testify truly to God's power and divinity, this says nothing about how clearly or accurately we perceive those attributes, only that we do. We can look at nature and say "God is powerful, and he is divine (other than creation)" but in reality that is as far as we can go. One can say "look at the orderly nature of creation, the Creator must be orderly." This is a true statement, however the reasoning is faulty. First of all, there are large portions of Nature that one could look at and say "This is not orderly." We see this play out in the so-called "Fine Tunning" argument. (to be transparent, the Fine Tunning argument specifically deal with the idea that the physical constants allow for life, not necessarily at every place in the universe. Nevertheless, I think that Tyson's critique of the argument that the universe, or even just Earth, appears to be designed for human life is legitimate) People look at the "orderly" nature of the universe and say "Only an orderly God could have done this." However, astrophysicist Neil Tyson points out that in all corners of the universe (including Earth) consistently are inhospitable to order and life. By this same logic I could say that since nature is cruel and nearly all life must kill other life to survive, that God is a god who believes that the strong should kill and eat the weak (and sadly, some people have made this argument utilizing Natural Theology).
Second: Human reasoning is fallen. Even if Nature were able to accurately and conclusively reveal details about God, we are so screwed up that we couldn't get to those truths. The Bible teaches that our hearts (which was the seat of human thinking for the Hebrew Mind) are "deceitful above all things and desperately sick (Jeremiah 17:9)." It also teaches that we were fallen from the very moment of our conception (Psalm 51:5), meaning that at no point in the development of human life are we able to reason unaffected by our sinful estate. The Apostle Paul goes so far as to say, quoting Psalm 14:1-3, that "None is righteous, no, not one; no one understands; no one seeks for God (Romans 3:10b-11)." If there is never a point in our lives in which our hearts and thoughts do not deceive us, and there is no one who is exempt from this estate... then how could we ever trust our fallen reasoning to approach a fallen creation to tell us about a transcendent and wholly other God?
Third: Natural Theology usurps Sola Scriptura or Scripture is Sufficient. Natural Theology often pushes us beyond the boundaries of what Scripture reveals. It makes statements about the nature of God that are not clearly spoken of in Scripture, one way or the other. Now, there is nothing wrong with this per se, however it plays out that Scripture is often interpreted in light of these conclusions. For example, philosopher William Lane Craig holds to a so-called Molinist understanding of God's foreknowledge. Without getting into a lot of details, this means that God has exhaustive knowledge of not only what is, but also what could logically be. However, this system requires God to be bound by what is logically possible, and this leads Craig to state "The counterfactuals of creaturely freedom which confront Him are outside His control. He has to play with the hand He has been dealt." All of this, based on a philosophical system constructed by fallen creatures, with fallen logic, observing a fallen creation. Let me say that again... Fallen creatures, with fallen logic, observing a fallen creation have stated that creaturely freedom creates circumstances that are "outside of his control." This sure doesn't seem like the God who "does all that he pleases (Psalm 115:3b)." God has given us the Scriptures as a sufficient revelation for all things concerning faith and salvation. This includes all of Theology. To affirm Natural Theology is to say that God has also given us another source of revelation that is sufficient concerning faith and salvation, and often times is to say that there are aspects of God's very nature or salvation that he has not revealed to us in scripture. This runs in contrast to the very heart of Sola Scriptura.
All of these things combined lead me to affirm the conclusion that the prolific 20th century Church Father Karl Barth did in his 1934 article denouncing Natural Theology, aptly titled Nein! To close, I leave you with a quote from British Theologian John Webster:
Once reason is thought of as 'natural' rather than as 'created' (or, to put it differently, once the category of 'the created' is collapsed into that of 'the natural'), then reason's contingency is set aside, and its sufficiency is exalted in detachment from the divine gift of truth. Or again, when reason is expounded as a natural competency, then it is no longer understood as fallen and in need of reconciliation to God. Again, when reason is considered as a human capacity for transcendence, then reason's continual dependence on the vivifying Spirit is laid to one side, for natural reason does not need to be made holy.
Christian theology, however, must beg to differ. It must beg to differ because the confession of the gospel by which theology governs its life requires it to say that humankind in its entirety, including reason, is enclosed within the history of sin and reconciliation. - Holiness, p. 11
Karl Barth and Emil Brunner, Natural Theology: Comprising "Nature and Grace" by Emil Brunner and the Reply "No!" by Karl Barth (Eugene: Wipf and Stock Publishers, 2002).
John B. Webster, Holiness (Grand Rapids: Wm B Eerdmans, 2003).