Is there a chemical or process that would dissolve or remove all other tissues in an organ (in my case, i would like a heart) and leave all the vasculature and blood vessels intact?
Yes, this can basically be accomplished via plastination, whereby they replace the normal water in tissue with a form of silicone. You can use it on virtually any kind of tissue; first time I walked into an anatomy lab I grabbed a large slice of something... about 1-inch thick, with a large hole in its center and began wondering what it was. Only when I turned it on its side and noticed hair and a tooth did I realize I was looking at a sagital cut of a complete head)
If you look up the Bodies art show, you can find some very nice examples of this (When I saw it several years ago, they even had the vascular tree of a complete body)
There is a procedure where you fill the coronary arteries and/or the venous system of the heart with acrylic resin, which quickly solidifies. You then submerge the whole heart into an acid solution which dissolves everything except the acrylic filling. What is left is a 3D replica of the heart arterial or venous system.