When I refer to the Freedom of Information Act (FOIA) I mean either the federal act or similarly implemented state laws. Clearly the validity of such requests would have to be limited to institutions which have received government aid in at least some way. In addition, there is the argument that such data could be considered a trade secret, as is the case in Mississippi.
However, the case in Mississippi is meant to protect findings themselves rather than raw data. In addition, there are many cases in which the data itself could be considered to have been public information to begin with, as is the case in archaeological excavation.
Some institutions already have an open data policy, such as PLOS, which requires all data relevant to a paper to be published:
PLOS strongly believes that, to best foster scientific progress, the underlying data from an article should be made freely available for researchers to use, wherever this is legal and ethical. Data availability allows validation, replication, reanalysis, new analysis, reinterpretation, or inclusion into meta-analyses, facilitates reproducibility of research and extends the value of the investment made in funding scientific research. Thus, PLOS believes that ensuring access to the underlying data should be an intrinsic part of the scientific publishing process. Furthermore, by getting data into the right place on publication we can reduce the burden on authors in unearthing old data, retaining old hard drives and answering email requests.
This is, at least in part, a purpose of FOIA: verification of claims. So based on the letter of the law, would such a FOIA request be considered valid and assuming the institution simply rejects the request, would there be legal recourse?