It's often said that the philosophy of mind "should be constrained by science".
However, much philosophy has always been constrained by science - at least to some degree. Straight after Thales, for example, Aristotle was both a philosopher and a scientist.
It's also been the other way around. Newton was imbued with the Platonism and the Aristotelianism his age reacted against. And many early-20th-century physicists (such as Mach, Poincaré and Einstein) were either Berkeleyan idealists or some kind of Kantian.
So the relation between science and philosophy has always been a reciprocal one.
Of course there are many philosophers around today who believe that because the mind and consciousness are so unlike the things which physicists or chemists study, that the philosophy of science must be an a priori pursuit – even an a priori science!
Although it's often said that science can’t tell a blind person what a colour looks like, an alien what a sponge feels like, or any human “what it’s like to be a bat”, there are still very many aspects of the mind that can be known by science. So perhaps the best thing to do is not to highlight the no-go areas of mind. Similarly, not to tell philosophers what are the acceptable areas of the study of mind.
As already hinted at, it can be said that science can't give us any information about "phenomenal consciousness" – or "what it is like" to smell a rose or listen to Mozart. Scientists, of whatever discipline, could tell us which neurophysical factors and features are causally responsible for all things phenomenal. They could/do even tell us which parts of the brain "subserve" mental events and consciousness generally. Though, some philosophers may argue, none of this has anything directly to do with the mind.
For example, a scientist may as well tell us that every time I form a mental image of the Cheshire Cat the light is on in my bedroom. Thus he could reduce my introspective image of such an image to the physical basis of the electric light and how that light impinges on my sensory receptors, then enters the nervous system… and, eventually, ends with a mental image of a Cheshire Cat. Even if it were the case than an electric light subserves - or were the "material substrate" of - my mental image, no neuroscientist could tell me anything about my mental image itself. Again, all they could do is tell us the physical substrate of such mental events or states - their "subveniance-bases". They could tell us no more than this.
However, it may still be the case that although science doesn't tell us what it's like to smell some horse manure or form a mental image of a cat, there are still many scientific factors which are relevant to these aspects of the philosophy of mind. And if not in these precise examples, then surely in other cases.
The question is:
What can science tell us about the mind?
Can science tell us anything about the mind?
Decades ago it became clear to many that we can't reduce the mind to the brain (as with the Identity Theory, etc.). It even became probable that no strict correlations between mental states (or events) and the brain could be found.
Other philosophers (such as Donald Davidson - with his "anomalous monism") argued that there are no mind-to-brain (or "psychophysical") laws; or, for that matter, brain-to-mind scientific laws. Not only that: there are no mental laws per se.
As for being constrained by science. This normative possibility doesn't make much prima facie sense. Philosophy has often been ahead of science. (And, indeed, vice versa.) So why should we constrain the non-empirical and non-experimental speculations of philosophy when so often in the past philosophy has shown science where it should go next? In addition, philosophers have also shown us which aspects of science have shown little logical or conceptual sophistication.
Though, of course, science simpliciter is itself speculative in nature. And scientific hypotheses have pushed science forward. So even if the philosophy of mind is still "aprioristic" (which, in most cases, it isn’t), then there would still be no good reason to place constraints on the philosophy of mind (or on any area of philosophy for that matter).
Neuroscience has told us many things about the brain. And cognitive science has also told us many things about the mind itself. Both areas have told us things about mental illness, "blind sight", the nature of colour vision, our cognitive faculties and which component parts of the brain subserve them, and so on. The science of psychology can also tell us about mental illness.
Here again psychologists don't need to know much about the neurochemical nature of the brain to empirically observe the behaviour of the mentally ill. Blind sight, at least in some cases, can be shown to be the result of brain damage or cognitive failures in the synaptic regions of the neurons. Colour vision is well studied by physiology and neuroscience.
However, and again, many philosophers still say that scientists can't tell us what the colour blue looks like or what it's like to suffer the pain of diarrhea.