The first worry is, as Bernard Williams put it, "reflection can destroy knowledge." The problem is that reflection might lead us to doubt an entire framework within which we had (what seemed like) knowledge. But Williams' point is more severe than that reflection can destroy appearances. (In some cases, that might be a good thing!) Rather, Williams suggests that if we allow that a person who has a set of value (or virtue) concepts which she can correctly identify when they are displayed by others (or in the world), and thus can correctly apply her concepts, then she has a certain kind of knowledge. (She knows when things satisfy the value-concepts she embraces.) But if she reflects on those concepts and comes to doubt their objectivity (say), then that knowledge becomes tainted: what she does with her concepts comes to look like one game among many. Her knowledge has been destroyed.
Mill became aware of a similar problem during his mental crisis. He questioned all of his commitments and found that his belief in them was easily weakened by his intrusive self-doubts. Upon recovery, Mill had this to say about reflection (and happiness):
Those only are happy (I thought) who have their minds fixed on some object other than their own happiness; on the happiness of others, on the improvement of mankind, even on some art or pursuit, followed not as a means, but as itself an ideal end. Aiming thus at something else, they find happiness by the way. The enjoyments of life (such was now my theory) are sufficient to make it a pleasant thing, when they are taken en passant, without being made a principal object. Once make them so, and they are immediately felt to be insufficient. They will not bear a scrutinizing examination. Ask yourself whether you are happy, and you cease to be so. The only chance is to treat, not happiness, but some end external to it, as the purpose of life. Let your self-consciousness, your scrutiny, your self-interrogation, exhaust themselves on that; and if otherwise fortunately circumstanced you will inhale happiness with the air you breathe, without dwelling on it or thinking about it, without either forestalling it in imagination, ot putting it to flight by fatal questioning. This theory now became the basis of my philosophy of life. (emphasis added)So, it seems, don't think too hard! I've always had doubts about this negative view of reflection, even though it is true that we can raise doubts about our own lives that are extremely hard to answer.
A different worry about the reflective requirement is that it doesn't (or shouldn't) apply equally to all of our convictions. Should I really reflect on whether slavery is wrong? Do I need to examine that?
I tempted to say that it depends on what we mean by reflection here. It seems right that I shouldn't doubt my conviction that slavery is (absolutely) wrong. But that doesn't rule out there being a point to reflecting upon why. Of course, the problem here is that at some point, we reach moral convictions that are bedrock, and the "why" question no longer has an answer. But if that's because the why question no longer makes sense, then I don't think reflection is destroying anything. Reflection might show us that the bedrock lies much deeper below the surface than we thought--that many of our convictions involve taking much for granted. Optimistically, reflection can help us better understand the structure (and relations between) our beliefs. (And here, I don't want to rule out that reflection can be informed by outside sources, e.g. by findings in moral psychology, etc. I can (and sometimes should!) bring empirical data to the armchair!) A related worry has to do with confirmation bias: that reflection is just going to give me whatever answer I want. However, this is a place where knowing the risks (and flaws) of reflection can make reflection better, even if not perfect. (We can't demand perfection.)
The last worry is that too much reflection will lead to inaction. We don't want to end up starving like Buridan's Ass. But sometimes a moment of "reflective paralysis" is a good thing: it stops us from doing something foolish, or dreadful. Equally important as reflecting on our convictions themselves is reflecting upon what we are prepared to do in the service of our convictions. (See, e.g., this previous post on tolerance.) This is another place where reflection might uproot other assumptions/convictions. (For example, my conviction that X is wrong is a different animal from my views about what should be done about people who do X.) Surely it's not a bad thing to reflect upon what we're prepared to do in the service of our convictions, and to consider whether that itself can be justified. (Or on the other hand, whether we're really not living up to our convictions...)