●Acute or delayed hemolytic reactions
If the blood type does not match your own (or is not compatible for some other reason) your body will attack it as if it were a foreign invader, because it is. This may happen within hours (acute) or it may take weeks to develop (delayed). Once the blood has been infused into your system, you are not able to separate your own blood from the injected blood without extensive medical intervention, and as such there's no easy way for your immune system to remove the foreign blood. You may become extremely sick as your body attacks your red blood cells and attempts to filter out the foreign blood, possibly leading to liver and kidney failure (and potentially death) without proper medical attention.
●Whole blood transfusion is becoming rare these days. Instead apherisis is used. The blood is taken into a separator, the required component is separated out and rest is passed back into the donor. Even blood donated to blood banks undergo this separation. For example, a dengue patient gets only platelets, person with minimal blood loss gets the plasma etc.
Red blood cells lack a nucleus and hence do not contain any DNA. White blood cells do contain the DNA of the donor. In cases where white blood cells are transfused, the donor's DNA does gets into the receipient blood. Since white cells has a finite life, these cells dies off in the due course and the DNA also goes off.