Heinz Frankel-Conrat (tobacco mosaic virus) wrote a series of at least 7 papers in the late 40s detailing the reactions of formalin with proteins. One gathers that there is a broad array of reactions, and any blunt attempt to reverse, for example, the condensation reactions, would also break biologically relevant molecules. And of course similar problems would occur for lipids, carbohydrates, and nucleic acids. And, in the usual case, the problem is further complicated by paraffin embedding.

That said, illumina's methods are certainly used on FFPE tissue, as is MALDI mass spec imaging. I can imagine protein crystallography might be absolutely contraindicated. What others?