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@ScottMorrison This is a very old comment, still to anyone who is wondering whether there are examples of a different nature they should check out the EHP spectral sequence, a spectral sequence which is constructed as an exact couple without any underlying filtered object at all.
I would like to add that I gave $MU$ instead of the simpler $H\mathbb{Z}$ because its also not true that localization w.r.t. all the morava $K$-theories (i.e. the harmonic localization) is the same as $MU$-localization which indicates in some sense that there are baziilions of non-trivial loclization of this sort.
