No the DNA does not change. The DNA cannot change unless there is genetic engineering happening like CRISPR. (Or random mutations)

There are things called retrotransposons (LINEs) that are able to move DNA but these are rarely expressed and as far as I know they are only intercellular and don’t move between cells.

But think about it, in an organ there’s hundreds of thousands to millions of cells each with their genome. The genome can not change on this scale, it is evolutionarily disadvantageous for a genome to be able to change that much (could lead to cancers and overall disfigurement of the proper transcription profile that the specific cell needs to function).

As others have said, organ transplant recipients need to take immunosuppressive drugs because the immune system recognizes the organ is not self. Doctors try to match the HLA profiles (the main receptors immune cells use to recognize self) so that as little rejection happens as possible, but it’s still not enough in many cases.