In a single normal individual human, all cells come from a single embryonic cell, and theoretically have the same DNA. However, there are 3 billion base pairs in the DNA of a single human, and even though the chance of mutation for any cell division is very, very small, there is a non-zero chance of mutation any time the DNA is replicated. So it is possible comparing any two cells in a single individual that you may find differences in the DNA on the order of parts per billion.
Specialization of cells for various systems can change the DNA content for that cell. For example, germ cell lines (sperm and egg cells) only contain half of the nuclear DNA of a normal cell. Red blood cells have no nucleus, and have none of the nuclear DNA of a normal cell. Mitochondrial DNA is a completely different issue. And in certain specialized cells, mechanisms are in place which prevent certain parts of the DNA from being accessible to transcription, locking out parts of the genetic code from access by the cell machinery. This is one difference between stem cells and non-stem cells - both contain the same DNA, but in non-stem cells, parts of the DNA are locked out.
If the mutations that make a cell cancerous were present in the embryo, the embryo would be non-viable. Therefore, for cancer to occur, there must be mutations in the cell after the individual is born. That isn't to say that there aren't mutations in embryos which lead to cancer, but they are almost always mutations in the error-correcting mechanisms of the cell and their true effect is to make it much more likely that subsequent cell divisions will produce mutations. It is the accumulation of mutations and mutations in critical genes which cause a cell to become cancerous. Most mutations in critical genes simply make it impossible for a cell to survive and it never reproduces, but very, very occasionally, a mutation in a critical gene will cause the cell to break the normal controls in the system on cell replication. At this point, the cell production increases, and tumors form.
With all of that as background, let me try to answer your question. Will two types of cancer have the same DNA? If they come from two different individuals, unless both individuals formed from the same original embryo (identical twins), absolutely not. Even normal cells from two different individuals will not have the same DNA. The differences may be very tiny, but they are there.
What about two types of cancer in the same individual? There are two different ways this might happen. One is that there is an inherent genetic defect in the error-correcting mechanisms of dividing cells in an individual. In this case, spontaneous tumors can arise in several places throughout the body. All of those tumors will be different, however, because the errors in cell division are for the most part random - it will simply result in a shotgun approach to knock out critical genes that prevent unregulated cell division.
The other way an individual might have two different cancers is that one cancer formed first and then metastisized. This requires mutations in several parts of the genetic code, knocking out both the regulation of cell division and the adhesions that cells form to each other. This typically happens because one of the earlier mutations in the cancer was a mutation that knocked out one of the error-correcting mechanisms. Again, the chances are that mutations are accumulating far more rapidly during cancer cell divisions than in normal cell divisions, and the metastasized tumor, having probably started from a uniquely mutated tumor cell, will have a (very very slightly) different DNA than the original tumor.
The short answer - no, even within a single tumor there are almost always genetic differences between tumor cells, simply because in most cases the error-regulating processes are broken and errors are accumulating with each cell division. However, these differences are on the order of a few changes in 3 billion base pairs, which is an incredibly small difference even compared to normal differences between non-identical twins. The DNA of an individuals lung cancer, for example, would be far, far more similar to the DNA of that person's normal cells than it would be to the lung cancer of another individual any less closely related than an identical twin.