The final couplet of the chapter is the one that has caused the most trouble, and the trouble is worth understanding because it touches on how the book was later read.
Not losing your place is to last. To die and not be forgotten is long life. The second clause puts death and long life in the same sentence, and the obvious reading is that what endures is not the person but the memory or the influence, which would make the phrase a modest one about legacy.
That is not how it was always taken. Later Daoist religious traditions were seriously interested in physical longevity and in transcending death, and they read this book as a source for those pursuits. In that frame the line invites a stronger reading, one in which dying and not perishing describes an actual transformation rather than a reputation. The Chinese permits both, since the character rendered forgotten in the ordinary reading can also carry the sense of being lost or perishing.
This is one of the clearest places where the philosophical and the religious readings of the text diverge on a single line, and the divergence is not a matter of one side misreading. The book is old enough and compressed enough that a movement forming centuries later could find its own concerns already in the words.
The chapter as a whole is neutral on the question. It offers a series of contrasts between the outward and the inward, and this last one contrasts holding a position with something that survives the loss of one. What that something is, it does not say.