I'm a bioengineer. Everyone that gives an answer mostly ignores epigenetics and the fact that RNA can permanently alter the _shape_ (or conformation) of how your DNA is tightly wound up. For example, RNA can direct methylation of DNA and alter histones, which can lead to transgenerational epigenetic effects on gene expression and phenotype [1]. The fact is, molecular cell biology is incredibly complex and the models we have are just that, models. Saying that these mRNA strands don't affect the genome long-term may be correct, but this is an educated guess based on theoretical models. There's a reason why FDA approvals traditionally took 10+ years, we normally verify our educated guesses empirically.

It almost certainly won't have long-term affects, but it may not be trivial to identify if mRNA vaccines have been altering epigenetics.

[1] https://en.wikipedia.org/wiki/RNA-directed_DNA_methylation