How are cell death like apoptosis and entropy related?
Unless you have a closed system, don't bother with the second law. These sound like some poetic descriptions but they won't tell you much about biology.


  It is argued that cell senescence is due to the accumulation of defects and wastes, but cell death could have a more active implication of carrying away the debt caused by utilizing information.


The first part of sentence which I italicized here is sufficient to understand the biology: cells age and die off because they accumulate defects. Organisms, whether single-celled or multi-celled, ultimately get around this by making lots of copies of themselves and expecting some to fail.

The second part of that sentence says "could" and is poetry to describe the first half, as well as the sentence that follows. It doesn't add any more science, unless you care about the existence of "entropic altruism" in the universe, which sounds to me like a religious rather than scientific approach. It's just stating that cell death (and perhaps, to an extension, death of multicellular organisms) can be useful to organisms/species rather than merely destructive. There is nothing wrong with finding beauty in physics and mathematics, but beauty in itself says nothing about how things work or how things should be.

In the context of a multicellular organism, cells kill themselves to not kill/harm the whole organism (e.g., to not develop into cancer, or as a planned event during development). The biological reason they do this is because multicellular organisms that have evolved genetic mechanisms to identify potentially harmful cells and have those cells die are more likely to reproduce and produce more multicellular organisms like themselves. If you want to interpret this as "entropic altruism" because cell death increases entropy, that's fine, but no thermodynamic argument is needed or even all that helpful in explaining the biology.

Note also that the book you cite doesn't seem to be in common use; it has no reviews on Amazon or Google books, and a previous book by the same author only has 1 review. I don't have anything against the author and they could be an excellent statistical physicist, but language like this makes me think they are not a biologist.