There are a number of ways for organic material to fossilize, the most common is by replacement and that is the one everyone is aware of. However, under certain conditions, a type of fossilization called permineralization can occur where the pores and voids are mineralized but some material remains bone. In most cases of this, depending on length of time the fossil is unexposed, even the remaining bone is eventually replaced. With something as recent as North American Equids and Equines many fossils are simply permineralized.
If you are interested, Google on 'Taphonomy'.
Did some looking around and didn't find anything all that convincing(to this layman). Fact is, I didn't find an article in my brief search that attributed any DNA from "permineralized" fossils. From this(one of the less academized) article they seem to imply that amber and similar substances which hermetically seal the future fossil(shades of Jurassic Park), is about the only way to retain any DNA over long periods of time.
Amber and Copal:
In amber and copal specimens, an organism becomes entrapped over a relatively short period of time in the various plant resins [exuded from trees such as Agathis, Araucaria (Araucariaceae), Bursera, Protium (Burseraceae), Hymenaea (Fabaceae), Liquidambar (Hamamelidaceae), Pinus (Pinaceae), Shorea (Dipterocarpaceae), and various Taxodiaceae-see Langenheim (1969) for a detailed breakdown of plant sources] that form copal and amber [Schlüter (1990) and Henwood (1992a) give detailed accounts of the imbedding process for specimens in amber and Pike (1993) and Henwood (1993) discuss taphonomy and collecting bias]. Though what is found in these amber and copal media is the actual specimen that died from thousands to millions of years ago, in many specimens, some amount bacterial action may have taken place over time and membranous material supporting the exoskeletal sclerites is often missing. Thus, when one attempts to recover the entrapped specimen by dissolving the amber or copal in a solvent, the result is often a disappointing slurry of floating chitinous plates. However, in some specimens preserved in this fashion, little or no decay has taken place and some muscle tissue complete with mitochondria survives and DNA can still be recovered (Henwood, 1992b). This had led to the exciting field of paleomolecular entomology in which the DNA of fossil organisms over 100 million years old can be studied and comparisons made with putative extant relatives.
Also, at the end of this same article, this item(specifically referring to insects???):
Other Forms of Preservation or Recovery:
Some forms of preservation of fossil insects do not encounter deterioration or mineral substitution of any kind. Those listed in this catalogue include subfossil and tar pit or brea recovery. The ages of these types of fossils listed in this catalogue are relatively young (most from the Pleistocene or Pliocene). In these fossils, all or parts of the insect specimen are visible in three dimensions much the same as those imbedded in amber or copal. Subfossil recoveries of Diptera have been made in peat bogs, caves, and ancient middens of small mammals.
Now if that says what I think it says, THESE specimens should have every cell intact. Which would lead one to believe the DNA would also be intact??? Color me confused.