No. The mRNA enters the cell but NOT the nucleus, were your DNA lives. It gives our cells instructions for how to make a harmless protein that is unique to SARS-CoV-2. It lasts just long enough to teach the cell how to encode the protein, and then the mRNA is destroyed. Then, if you’re subsequently infected, your body recognizes the virus should not be there and the immune response is triggered to fight it off effectively.
This is an amazing technology that has been in development for many years, but was able to be accelerated due to everyone’s focus during the pandemic.