Could research data fall under the Freedom of Information Act? 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?
The original Freedom of Information Act applied only to information in the possession of government agencies. That means that the release of reports and proposals submitted to the funding agency could be compelled under the FOIA. But it did not apply to data from the research, if in possession of the PI and not the funding agency (as is typically the case).

A later amendment (Shelby amendment) expanded the FOIA to apply to some federally funded research data in the possession of a non-profit institution, as follows:


The release of "research data relating to published research findings produced under an award that were used by the Federal Government in developing an agency action that has the force and effect of law" may be compelled under the FOIA, if
the data is not "trade secrets, commercial information, materials
necessary to be held confidential by a researcher until they are published, or similar information which is protected under law."


Also, the requestor may be charged a "reasonable fee equaling the full
incremental cost of the agency, the recipient, and applicable subrecipients."

So the general answer to your question is that in most cases, the FOIA does not compel the release of data from federally funded research.

Regarding state law, it seems from the appendix to this report mentioned in another answer that at least one state does have an open records statute that may apply to research data produced by employees of the state university system:


  "Measuring Reproducibility in Computer Systems Research." Christian Collberg, Todd Proebsting, Gina Moraila, Akash Shankaran, Zuoming Shi, Alex M Warren. March 21, 2014.