While most, or at least many, Christians believe that healing by faith still happens today to some extent, this goes beyond that — to a belief that visiting the doctor at all is somehow wrong, or a lack of faith. So it's not just a question of "Can faith healing work", but "What prevents you from doing something else also?".
The simple answer might be that seeking other treatment is a sign of doubt, and could therefore undermine the faith portion of faith healing, but I find that explanation unsatisfying, from both a biblical perspective and from the eyes of a parent. By not visiting the doctor, I would only be denying God an agent He might use to either effect the healing or show his power.
There is a better explanation from biblical study. It comes to an interpretation of how we understand the silence of scripture (things that are not explicitly discussed), and whether silence is permissive or prohibitive. The bible never explicitly permits or prohibits visiting a doctor, so how are you to decide on this? In the general case (not just scripture), we often think of silence as permissive. However, it is easy to show that this is not always true. I'll provide an example:
Let's say you send your teenage child to the store for milk and eggs, and give him/her a $20 bill to cover the expense. The child comes back with milk, eggs, ice cream, and less change than you hoped for. You didn't tell him/her not to buy the ice cream, but you're probably not going to be happy. Here, the silence is prohibitive. You also didn't specify which store to go to. You may have expected the local grocery store, but if the child knew of a sale at a nearer convenience store they may have chosen that instead. Here, the silence is permissive.
Applied to scripture, there is a common (not universal) interpretation that when we are given instructions in an area, silence on other aspects of the area is prohibitive. For example, God told Noah to use gopher wood when building the ark. He didn't mention other kinds of wood (he was silent), but his specific instruction for gopher wood excludes/prohibited them. What if Noah wanted to use gopher wood for the hull, but thought a little bit of maple trim might have looked nice around the window and door? This interpretation system would not allow for that. Going back to the shopping example, ice cream was not on the list, and so was prohibited.
Why is all this relevant to the faith healing question? Well, it just so happens that the bible is not entirely silent on the subject. We are told to "pray for the sick and annoint them with oil." (James 5:13-16). Using the above interpretation system, this excludes the use of doctors and medicines.
It comes down to hermeneutics. Hermeneutics are the lens you use to study scripture. As an example, they are how you know what to do when you come upon "Do not judge, or you too will be judged." in one place, and "Do you not know that the Lord’s people will judge the world?" in another. Hermeneutics help ensure that you interpret scripture consistently; that you not hear only what you want to hear from the passages you want to read. A perfect hermeneutic, perfectly applied, would lead to a perfect understanding of scripture. Unfortunately, man is far from perfect.
Most good hermeneutics will at some point need to deal with the silence of scripture, in some way or another. Most fail to handle it adequately. The hermeneutic described above actually handles the subject pretty well, but hermeneutics are a creation of man, and so are not flawless or perfect. Our application of them also often leaves something to be desired. What we have in these tragic cases is a strict hermeneutic, strictly applied (neither of which is bad on it's own), and a lapse in judgement about it.
That said, if you're studying the bible without a good hermeneutic close at hand, you're probably not getting what you should out of your study. Sadly, this is all too common. It's just important to remember that the hermeneutic is a starting point or measuring rod for your interpretation. It's not the whole interpretation, but one tool you should use.
