Getting a standard blood transfusion cannot and will not change your DNA. In a blood transfusion, the donor’s red blood cells or blood plasma are given to the recipient, neither of which contains any genetic material (red blood cells lack a nucleus, which is the part of the cell that contains the DNA). White blood cells, which contain the DNA, are filtered out from the donor’s blood prior to the transfusion.

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.
There are some transfusion procedures that 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.
Overall:
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.