Animals don't really smell DNA or any nucleic acid
I am not aware and really truly doubt any mammalian (nor vertebrate/invertebrate) organism can detect DNA/RNA via the olfactory system (nose, antenna, sensilla, etc.).
Typically we think of the olfactory system as reserved for volatiles (odors), and the gustatory system for tastants like salt and sugar and bitter compounds. A dog cannot smell DNA as typically present in nature in ecologically-relevant contexts, as DNA is not volatile at all (it's a giant molecule, and things you can smell are typically very small in molecular weight, with a low vapor pressure). Even if present in droplets, it wouldn't be detected by olfactory receptors, which are the interface responsible for transducing chemical "odor information" into electrical information processed by the nervous system. I once also answered a question about whether one can smell or taste viruses and spores; the answer is categorically no. You can only smell or taste some of its components can can become airborne. You could perhaps smell some outcome of a viral infection or the metabolites or by-products of mold, but not the cells or viral particles themselves.
Can DNA be incorporated in the nasal/olfactory epithelium?
Oh, no. It's even less likely is the possibility that DNA is taken up by cells in the olfactory epithelium. And even if it was to be taken up by cells, genome incorporation of random DNA is pretty much impossible. That's extremely unlikely, on top of extremely unlikely.
However, can we taste nucleic acids like DNA?
You can perform this test yourself at home! The easiest way would be to try isolating some DNA from strawberries, which are octaploid and a great source of easy DNA! If you were to taste it, according to one person, you would find it
salty and slimy after sitting in his mouth for as long as he could
stand it. Conclusion: strawberry DNA is best eaten accompanied by the
rest of the strawberry.
Probably salty due to the sodium and potassium ions that neutralize the charge of the phosphate backbone that is present in DNA.