Let A be any non-zero abelian group, let C0 be the group of (infinite) sequences of elements of A, and let C1 be the subgroup consisting of those sequences that are almost-everywhere zero. Then C• is a bounded chain complex with non-zero homology, concentrated in degree 0; the homology group H0(C) is an example of a filterpower (of A), a generalisation of ultrapowers.
Now, consider the (ascending, integer-indexed) filtration F•C• where F–mC0 is the subgroup of sequences whose first m entries are zero and F–mC1 = F–mC0 ∩ C1. This is an exhaustive, separated filtration, albeit unbounded below. It is not hard to see that the inclusions F-(m + 1)C• → F–mC• are all quasi-isomorphisms, so the associated graded chain complex must be acyclic.
In particular, the E2 page of the homology spectral sequence associated with F•C• is zero! But this should not be surprising; after all, the induced filtration on H∗(C) is trivial. One can make various modifications to this example, but the essential point is that the filtration on H∗(C) can fail to be separable even if F•C• is separable.