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A recent example is the question of the existence of outer automorphisms of the Calkin algebra of a separable infinite-dimensional Hilbert space (this is the algebra of all bounded operators divided by the compacts). See this paper of Farah for details.

A feature this example shares with Kaplansky's conjecture mentioned above is the use of CH for the construction of the desired object. For the other direction (proving the non-existence of an outer automorphism), which is more interesting, Farah uses a natural combinatorial axiom (OCA) which can be forced in any model of ZFC.

Wikipedia also has a list of independence results.(I cannot link to it though because the system doesn't allow more than one link per answer).