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!
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It sort of does exist, see Mitochondrial DNA
– Scott Downey – 2016-01-11T15:23:06.3202I don't really understand what you mean. DNA is the support of genes. Genes are more or less independent and fulfil a specific task. My thinking is that if we parse a single DNA into genes, there would be no differences with the parsing of multiple DNA into genes. More or less like a multidimensional array can be casted into a single-dimensional array. – Kii – 2016-01-11T15:32:01.050
@ScottDowney mtDNA is a (small) part of the same DNA as the rest of the body, not a unique DNA set. I'm interested in the latter. – Frostfyre – 2016-01-11T15:40:00.973
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This is a recent case that you may be interested in : http://www.independent.co.uk/news/science/human-chimera-man-fails-paternity-test-because-genes-in-his-saliva-are-different-to-those-in-sperm-a6707466.html
– Kii – 2016-01-11T15:49:57.6831@Kii I think I heard about that. Truth is stranger than fiction. – Frostfyre – 2016-01-11T15:52:39.750
7@Frostfyre: AT the scales you're talking about the distinction between 'the same' and 'different' gets a little blurry. Viral DNA strands replicate using human cellular structures, but they aren't human. Mitochondrial DNA is noticeably different from other DNA used elsewhere, to the point where it's posited that mitochondria were actually consumed by other single celled organisms before undergoing a (now very, very, long term) symbiotic partnering. – Joe Bloggs – 2016-01-11T16:03:42.250
1@Frostfyre are you talking about fundamentally different types of DNA based on different chemistry? – Scott Downey – 2016-01-11T16:23:59.087
@ScottDowney Not different chemistry (though that would be interesting), but different nucleotide chains. – Frostfyre – 2016-01-11T17:33:29.407
3There needs to be some way for all the different sets of DNA to make it into the offspring. Otherwise, no problem, as the answers demonstrate. But do think about what reproduction would look like. Also, there are probably hundreds to low thousands of independent genetic regulatory networks in humans, which is pretty much a complex version of what you're suggesting. There just isn't any good reason to keep the DNA segregated (especially since the networks partially overlap), so it's all run off the same DNA. Chromosome condensation is used, in part, to control which subset is used in each cell. – Rex Kerr – 2016-01-11T22:58:39.887
They are discovering more and more alternate DNA inside you all the time. Humans are messier than you think. I think your idea is completely plausible within current knowledge. Paywalled, I'm afraid, but might give you something to search on: https://www.newscientist.com/article/mg22930550-400-strangers-within-meet-the-other-humans-who-live-in-your-body/
– Whelkaholism – 2016-01-12T17:18:04.670@RexKerr If there were a "Master" set of DNA in the reproductive cells (ovaries, testes, and the ova and sperm they create) and in undifferentiated stem cells, but then as the cells begin to differentiate, certain chromosome pairs could simply be dropped entirely. Say one chromosome pair had code for making eyes. It would not be needed anywhere else in the body, so it could be dropped from everything but eyes and gonads. The question is whether the savings from not bothering to replicate that chromosome pair is enough to justify the mechanism to govern it. – Monty Harder – 2016-01-13T18:25:11.777
@MontyHarder - Indeed. Some organisms do that in some tissues, to some extent, already. (Oxytricha, for instance, or https://en.wikipedia.org/wiki/Polytene_chromosome (which is admittedly not as specific as you mentioned).) Note that there is tons of reuse in biology, though, so the "not needed" bit is mostly not actually the case.
– Rex Kerr – 2016-01-13T19:31:23.007