Well, this question could refer to any cell type in the body, but blood provides the easiest example.

 

In red blood cell production, erythropoiesis, if the source of this production (certain bone marrow from what I gather) was replaced with a cell with a different genes (either genetically modified cells or cells from a different organism), would the new blood cells' DNA eventually come to be the most common in the body? If so, what long-term effects would this have on the organism?

Blood cells have no DNA. They're made from certain expressions in our genome being translated into proteins. Once the blood cells are assembled, they do not require DNA to carry out any of their functions.

Blood cells have no DNA. They're made from certain expressions in our genome being translated into proteins. Once the blood cells are assembled, they do not require DNA to carry out any of their functions.

that's almost true... mammal red blood cells lack a nucleus and DNA, but white blood cells and the RBC of other animals have them.

In answer to the question about other types of cells, I'm going to say no, since you're essentially just talking about an organ transplant.