The issue of whether delusions are beliefs has been central to philosophical work on delusion, as several of the previous posts here reflect (see especially Bortolotti and Gerrans). I'd like to express a few reactions to this debate.
Obviously, before we can ask whether delusions are beliefs, we need to get clear about the nature of delusions, and the nature of beliefs.
Let's start with delusions. Delusional behaviours, as well as the conditions that give rise to them, are extremely varied. Only scratching the surface of this variety, one can point to the difference between the delusions that occur in the context of localised brain damage, and those that occur in the context of schizophrenia. The former tend to be monothematic and circumscribed; the latter, polythematic and elaborated. In the former, the content of the delusion (what it's about) is often explained in terms of localised damage to dedicated regions or processing streams - so delusional misidentification is hypothesised to result from damage to parts of the brain dedicated to, e.g., emotional reactions to faces (see Ellis and Young 1990), or the tracking of individuals (see myforthcoming paper in Philosophical Psychology). In the latter, the delusions are taken to be the result of more global changes in experience, and the content tends to be more varied, florid, and congruent with a global delusional mood of paranoia or unreality (e.g. Jaspers' General Psychopathology, 1963). There are more differences between these two kinds of delusional phenomena, and, there are many more than just these two kinds. My first reaction, therefore, is concerned with variety. Philosophers should generally be more sensitive to the variety of cases that there are. So, when asking the question, “Are delusions beliefs?”, one should be prepared to have to say: “Some are” and “Some aren't”, or, indeed, to propose a spectrum between the more or less belief-like. (Unless, of course, the very features that make something a delusion are the same features that rob it of belief-status, but that is a very strong claim).
What about belief? I clearly don't have space to adequately discuss what is perhaps the central concept in philosophy of mind. For now, suppose that what a subject believes is how the subject takes things to stand in the world. Coupled with certain motivations, this will dispose the subject to act (not merely behave) in a certain way. In light of this view, why would someone want to deny belief-status to some delusions? That delusions are formed on the basis of poor evidence and are resistant to correction, will not, on this view, rob them of belief-status. As Bortolotti rightly points out, healthy people do seem to believe things tenaciously and contrary to evidence, and these beliefs are often reflected in their dispositions to action. That epistemic irrationality cannot alone rob a state of belief-status is also reflected in the fact that “irrational belief” is not a contradiction-in-terms. What is perplexing about delusional patients is that they don't act in ways that are coherent with their delusional claims. The central question is not whether a mental state, the delusional state, is a belief-state or some other state, but rather, whether we can take the subject's delusional claims as an indication of how they really take things to stand in the world.
It is important to distinguish a question about what it is to believe from a question about how, in fact, we human beings do it. For me, these cases of incoherence (perplexing insouciance on the part of the Capgras patient or double-bookkeeping on the part of the schizophrenia patient) aren't interesting because they address the first question. They do not inform or threaten our philosophical concept of belief. Rather, they illustrate that being the sort of thing that is capable of having beliefs is a serious cognitive achievement. To look at it from another angle, instead of asking how you have to damage a believing system, so that it fails to be capable of believing, we can ask: how would you go about building a system capable of believing? What kinds of memory subsystems would you need? What kinds of inferential subsystems? What kinds of demands would need to be made on these? Here we get to some really interesting and substantive issues.