An active replicator is that which exerts at least some kind of influence over the likelihood of its being copied. A DNA molecule is a replicator; a piece of sheet music considered valuable enough to be xeroxed can be considered a replicator (Dawkins, p. 83); but an individual organism, sexual or asexual, cannot be considered a replicator.
Because we reject Lamarckism, we know that characteristics acquired during an organism's life cannot be passed down to its offspring. Even in the case of asexual reproduction in which the entire genome is passed down, any acquired features--a lost limb, developmental factors, etc.--is not replicated in the lineage, and in order for the organism as a whole to be called a replicator it must pass these on. "There is a causal arrow," writes Dawkins, "going from gene to bird, but none in the reverse direction." However, the genome of an organism that reproduces asexually could be a candidate for the term "replicator."
Can a species be considered a replicator? Again, Dawkins points out that a species, being mutable, is not comparable to a gene and its alleles. However, the gene pool of a reproductively isolated species could be a candidate.
In examining this possibility, Dawkins explores "differential lineage extinction"--for example, the statistical rates of extinction of ammonites and bivalves that have a high rate of evolving a larger size, say, than those who do not (those that more rapidly increase their size over successive generations are also more likely to die off). However, while these different rates of extinction are a form of selection, they do not drive "progressive evolutionary change" and thus these lineages are not replicators, either. They are merely, as Dawkins terms them, "survivors."
(And at this point, people, I confess I must read the chapter again.)
Part II: Punctuated Equilibrium and Memes