If an organ from person A is transplanted to a new human body B, is it possible that we can detect A's DNA in B?
How long until the organ's DNA is replaced by B's DNA so that we are no longer able to detect any signature of A?
DNA will be never replaced (unless you are speaking about something, where DNA might be only trash, like in the case of blood transplantation or in the case when "organ" would be slowly replaced itself by host regenerative power and "organ" transplantation would be only something temporary, but then we are probably speaking about wider definition of organs).
For further evidence, you may look at cases of chimerism. That is, when young embryos fuse together and create single one with two different cell-lines, which later differentiate to create different organs. https://en.wikipedia.org/wiki/Chimera_%28genetics%29
Furthermore, rather more common but with less drastic effect can be somatic chimeras, that is when (usually) early, when embryo is formed only by small amount of cells, one of cell has a mutation, then later in life, this cell again can give rise to organ or several of them. In that case, those organs will carry this mutation while rest of the body will not. This can be potentially more common as this can happen in any single time during development and even after it (in fact, cancer cells are sort of product of that). More interesting cases here: https://en.wikipedia.org/wiki/Mosaic_%28genetics%29
So, from simple fact that these cases exists and are well-spread, and basically means that several different cell-lines can live in the same body, the answer on your question, if organ DNA will be replaced after transplantation is no.
edit: In the case of whole cell replacement, they can be from both donor and host. However, there are a few problems:
Most of cells that we have are not totipotent, not even pluripotent or multipotent (see: https://en.wikipedia.org/wiki/Cell_potency), most of our cells are plain and stupid. This means that highly specialized cells of some organ would probably be replaced only by other highly-specialized cells of that organ (or their precursor cells that are kept for this solo reason, keep regenerativ ability of particular organ, as quite a lot of highly-specialized cells can't divide any more). However, there are a few documented cases with regnancy, where a few cells (embryogenic, thus with bigger "healing" power) migrated through placenta to mother's body and healed her diabetes. So theoretically this is possible, as well as host getting cancer from donor's organ. But again, reason why we are transplanting organs is because human body is not particularly great in regenerating itself and those organs won't grow. So theoretically possible, but not great concern (except for cancer).