No, in fact, mechanisms in cells are in places so that "heteroduplexes" (meaning one filament of RNA and one of DNA paired together) are quickly degraded.
Secondly, the RNA doesn't even need to get to the nucleus, where the DNA is. All it needs is to land in the cytoplasm, where it will get processed by the protein synthesis machinery (ribosomes in primis) to produce the antigen. RNA in the cell isn't that stable as well, so it will be ultimately get degraded.
See the section on COVID-19 near the end.
AFAICS, this is not BNT162b2, which is the one that we are talking about. There are 3 other candidates, one of them is the self-amplifying one.
The paper seems to imply maybe not? https://www.sciencedirect.com/science/article/pii/S187962572...