While the other answers have responded to the title of your question, I don't think anyone has really addressed the specific question you asked. Can an organism use different sets of DNA in different organs or organ systems? The answer is almost certainly yes, with a couple of caveats.
Caveat 1: The reproductive cells of the organism, the ones that give rise to all the other cells, will need a complete copy of every organ's genome, otherwise that organ's genome, since it doesn't reproduce itself, will be lost. This means that your embryo is going to have all the DNA needed to create every organ in the body, and as the cells differentiate they will discard the DNA they no longer use. The germ line cells however will retain all of the DNA for the next generation.
Caveat 2: Many of the genes in a multicellular organisms are actually used by every cell. These are called housekeeping genes and they handle things like basic upkeep of the cell. Genes like RNA Polymerase II, Ribosomal RNAs, and the Nuclear Pore Complex are going to need to be in every cell, so your different DNA sets are going to have a large amount of overlap.
That said, I see no reason why a multicellular organism couldn't handle differentiation via loss of DNA. Currently, all organisms I've ever heard of create different cell types and organs by differential regulation of the genes in their DNA. They have a million ways of modulating the output of genes and create incredibly complex regulatory networks full of feedback loops and bistable switches. By comparison just deleting the genes necessary for being a neuron when you decide to become a glial cell seems pretty straightforward.
The only trick is that the DNA for different functions needs to be spatially segregated in the genome for easy deletion. The simplest way would perhaps be by chromosome. For instance, say chromosomes 1, 2 and 3 contain all of the ubiquitous housekeeping genes, 4, 5, and 6 contain ectoderm specific genes, 7, 8 and 9, contain mesoderm factors, and 10, 11, and 12 are necessary for endoderm. When a cell decides to become ectoderm it simply degrades chromosomes 7-12.
Obviously its sort of tricky for this sort of spatial organization to have evolved. Multicellular organisms would have had to originally start off with this method of differentiation. If that's the case however, regulation of what cells do what actually gets a whole lot simpler since the cells really have no choice in the matter. On the upside, you've always cured cancer!