Getting a standard blood transfusion cannot and will not change your DNA. Most people only receive red cells or blood plasma during medical procedures, and neither one of those blood components contain any DNA material. Transfused blood still needs to be a match to the recipient's blood type, including the ABO blood groupings. A blood test performed after a standard blood transfusion would reveal only the patient's DNA profile.
This isn't to say that human blood does not contain any DNA, however. White blood cells, which are usually removed from donated blood by a centrifuge, do contain DNA. If someone were to require a whole blood transfusion, the donor's white cells would enter the recipient's bloodstream and remain there until they expire, generally within four to eight days. Such whole blood transfers are rare, however, and the donor's DNA would not survive long enough to have an effect on the recipient's DNA. Conceivably, a blood test taken shortly after a whole blood transfusion could show a mix of DNA coding, but not strictly the DNA of the donor.
An episode of the television series M*A*S*H dealt with a racist white soldier who specifically asked the doctors not to give him any blood from a black donor. In an effort to show the patient the error of his ways, the doctors used iodine to darken his skin. When the patient awoke, he discovered he had turned "black" as a result of a blood transfusion. The doctors revealed their ruse only after lecturing the patient on the realities of blood donations. Receiving a blood transfusion from a donor of a different race would not change the recipient's own genetics.
Another television series, Law and Order, presented an episode in which the prime suspect was initially exonerated by a DNA blood test. Blood drawn from the suspect's arm did not match the blood found at the scene of the crime. Only after the suspect died did the detectives discover what really occurred. The suspect had implanted a plastic tube containing another person's blood into his arm, and that foreign blood was used in the original DNA test. Had the blood entered the suspect's own bloodstream, the test would have revealed the true killer's DNA. The foreign blood had to be kept separate from the killer's own bloodstream.
There are some transfusion procedures which can change the recipient's DNA, however. Bone marrow transfusions, for example, often require that the recipient's own blood and marrow be destroyed in order to reduce the chances of rejection. Once the donated marrow begins producing red blood cells again, the white blood cells would most likely contain the DNA of the donor, not the recipient. This is why finding a close genetic match for bone marrow donation can be so vital.
Receiving a standard platelet, plasma or red cell blood transfusion will not change the recipient's DNA at all. Receiving a whole blood transfusion might skew the results of a DNA test for a few days, but eventually the recipient's own blood cells should overwhelm those of the donor. Only a systemic process such as bone marrow transfusion could actually change the DNA profile of a recipient.