No. The mRNA administered with the vaccination is unable to reach the nucleus and incorporate itself in the human genome (DNA/genes). Furthermore, both the mRNA from the vaccine and the proteins produced in the cells are quickly broken down by the body once more. This is because by this time the body has already launched the required immune response and the mRNA and the proteins are no longer needed to protect the body against the coronavirus.
The mRNA administered with the vaccine is not dangerous for the body. The human body produces mRNA naturally and requires it in order to make vital proteins. Incidentally, in case of an infection with the coronavirus, virus RNA also enters our bodies.
Even though the mRNA vaccine is based on a new technology, research into mRNA vaccines has already been going on around the world for around ten years. This means that a number of studies and results are already available.
All results on the efficacy and safety of the vaccines can be found on the Infovac website (available in German, French and Italian).
Find out more about the mRNA vaccine in the question: What is mRNA and how does an mRNA vaccine work?