Assuming that all environmental conditions on earth remain the same in
distant future, the tendency of nature to increase entropy would cause
the chemistry and the mechanism of DNA replication to create more and
more "errors". Could these errors accumulate over time for all life,
An accumulation of errors in replication over time can certainly be assumed and observed. However, these errors -also called evolution- are generally not related to entropy.
Entropy, as a statistical concept (Boltzmann), describes the population of different states, like heads or tail in a coin-flip. If you toss 100 coins, the outcome of 50 heads would correspond to the maximum entropy, while 0 or 100 heads would be the lowest possible entropy. Boltzmann also describes that higher temperatures equalize the population of different states. Too high and too low entropy would both mean death.
failure of replication by means known today, bringing life as we know
it to extinction?> In other words, will life eventually fail to continue reproducing due
to the inability to maintain its inherent state of non-equilibrium?
If an error leads to a decreased probability of reproduction, it dies out and will not accumulate. On the other hand, in the rare instances that the error actually increases the probability of reproduction, it has a high chance to accumulate. So in general, accumulation of errors would increase the ability of reproduction.
Not being able to maintain the correct state of entropy definitely is fatal and is the natural end of life when you grow old.