This is a very general answer without specifics as treatments do differ depending on patient etc., but gives you the basics.
Chemotherapy and radiotherapy both take advantage of the mechanisms by which cells reproduce to destroy the cancerous cells. Neither therapy alters your DNA.
Chemotherapy (generallyspeaking) impairs the division of fast-dividing cells (like cancer cells, but also those responsible for hair growth, hence the associated hair loss) thereby killing off the cancerous ones.
Radiotherapy does destroy the DNA of cells, but healthy cells have means for repair (the cancer cells often do not).
Obviously radiation exposure can cause cancer by causing mutations, however radiotherapy is applied in such a way as to give high doses to a tumour and limit the amount of exposure to surrounding cells (otherwise it wouldn't be an effective treatment).
So neither will fundamentally alter your DNA and the results of a test would not change. A DNA test does not generally look at your whole DNA, but will look for specific markers based on what the test is looking for e.g. markers from a father to determine the paternity of a child. The amount of markers in common determines the outcome of the test.
If you are receiving either (or any) treatment, the physician should explain all of this to you. If they don't, make sure you ask.
Hope that makes sense!