Field theories are nonlinear because the quantum fields satisfy nonlinear dynamical equations.
But renormalization does not make quantum fields into a nonlinear functional of test functions. The Wightman distributions are, by definition, linear functionals of the test functions, and Wightman distributions always encode renormalized fields.)
Instead it changes the space of test functions to one where the interacting quantum fields are perturbatively well-defined. This gives a family of representations of the field algebra depending on an energy scale. All these representations are equivalent, due to the renormalization group, and the corresponding Wightman functions are independent of the renormalization energy. (In simpler, exactly solvable toy examples that need infinite renormalization, this can actually be checked.)
The dependence on the energy scale would not be present if contributions to all ordered were summed up (though nobody has the slightest idea how to do this nonperturbative step). The energy scale is simply a redundant parameter the influences the approximations calculated by perturbation theory.
The renormalization group is an exact but unobservable symmetry (just like gauge symmetry) that removes this extra freedom, but as computations in a fixed gauge may spoil gauge-independence numerically, so computations at a fixed energy scale spoil renormalization group invariance numerically.
Note that Wightman functions are in principle observable. Indeed, the Kadanoff-Baym equations, the equations modeling high energy heavy ion collision experiments. are dynamical equations for the 2-particle Wightman functions and their ordered analoga.
[added 22.01.2018] In the above, the renormalization group refers to the group defined by Stúckelberg an Bogoliubov, not to that by Kadanoff and Wilson, which is only a semigroup. See here.
Best Answer
In my corner of things what comes to mind is a recent paper by Atiyah and Moore A Shifted View of Fundamental Physics.