Take any automorphism of a compact topological space X that has a non trivial action on homology, cohomology or any other homotopy invariant algebraic topological invariant.  Then the free group generated by the automorphism acts on X and so it acts on C^0(X). But clearly the Z action cannot be extended to an R action, because that R action must come from an R action on X [1]. Such an R action would be homotopically trivial contradicting that the generator of the Z action gives a non trivial algebraic topological invariant. 

Example: the action of the mappingclass group on a two dimensional oriented surface Sigma  which give non trivial elements of  Sp(2g,Z). In particular the so called Dehn twists, give examples with non trivial actions on H^1(Sigma). 

Now obviously this is not a non commutative example, but if I understand correctly the equivariant case (a topological space with a toplogical group action) gives rise to non commutative C^* actions that are related to the space with a a group action in much the same way as above, i.e look for a space with an equivariant automorphism that gives rise to a non trivial element in equivariant (co) homology or other equivariant invariants. 

[1]  The Gelfand Naimark theorem is not just that the statement about maximal ideals of commutative C^* algebras correspond to points, but also that continuous *-homomorphisms correspond to continuous maps. Its a functorial equivalence of categories!

[2]  the Z- rank of H^1(Sigma) is 2g which comes with non degenerate symplectic form defined by cup product and evaluation on the fundamental class

[Sigma] \in H_2(Sigma).