These are interesting questions.
I think DNA is considered pretty stable, but certainly it could degrade over time, especially depending on the environment it is in.
In biology, the DNA is usually contained to a cellular with presumably nice for the DNA conditions. In addition, the biological environment has DNA repair mechanisms. Its not clear if those mechanisms would work on the DNA in the conditions of the nano-arm. The DNA does not look like DNA packaged onto histones which is the way it is found in eukaryotic cells.
Perhaps the people working with these nano-machines will develop a controlled mini-environment (similar to a cellular environment) in which they would work, that would minimize damaging conditions.
Without the DNA reproducing, any damage to the DNA mechanism would probably result in a short term and possibly terminal (to the nano-mechanism's) malfunctioning, but not what I would call scary things (like infecting people). If the DNA were reproducing in some way, then you could have the possibility of sequence evolution occurring, with selection possibly generating things like viruses or other "pathogens" (pathogenic to the reproducing entities, not necessary normal biology, though that might be possible also).