Are there any drawbacks to having immutable DNA during lifetime?
Semantic of the post
This question could arguably be better on-topic on WorldBuilding.SE. The difficulty with the kind of question that ask What if the world was different, it is unclear how different would it exactly so to understand what consequence to draw from this scenario. The other issue is of course that the answer cannot be drawn from empirical studies but only be inspired from current knowledge and make untestable predictions.
Because definitions of epigenetic vary and some of them might be extraordinary broad, I will assume that by mutation, you are referring to any change in the DNA sequence excluding epigenetic changes.
Related post
Does our DNA change during our lives? is a related post.
What do somatic mutations do?
Adaptive Immunity
For species having some form of adaptive immunity, somatic mutations are used for producing a large diversity of antibody than can be tested on the self before being released in the body searching for antigen to bind to. In absence of somatic mutations, the adaptive immunity is pretty much inexistent. This point is probably the most important and has been raised by @Polisetty in the comments.
Cancer
Somatic mutations cause cancers. More generally speaking, somatic mutations create genetic diversity within the body yielding to opportunity for selfish cells that could proliferate to the expense of the rest of the body. In absence of somatic mutations, there would be no (or at least much fewer) cancer. This will affect selection for increased lifespan.
Somatic mutations on genetic diversity
As a fraction of the mutations that are transmitted to the offspring occur before the separation between the somatic lines and the germline, the absence of any somatic mutation would result in a lower mutation rate. This would result in lower genetic diversity overall. It would also affect selection pressure that depend upon the mutation rate and may have other consequences that are hard to predict.
Also, and more importantly, many species undergo asexual reproduction such as budding in plants. In some species, the entire genetic diversity is caused only by somatic mutations. In absence of such mutation, there would be no evolution possible.
What mutations are inherited
If a parent has a mutation in the lineage giving rise to half the gametes, then this mutation will necessarily be passed to half of the offspring. For this reason, offspring tend to share new mutations even if their parent did not have them. Absence of somatic mutations will remove this effect, which I would predict would have little impact on anything!