Quantum field theory in Minkowski spacetime has CPT symmetry, which is antiunitary (section II.5 in ), so it's not an inner automorphism. But it's not an outer autmorphism, either; it's not a (linear) automorphism at all. Since the question refers specifically to outer automorphisms, I'll try to address that. This is pushing the limits of things I've only barely begun to understand, so I'll lean heavily on the literature.
In the context of quantum field theory, in any given model, we can consider the von Neumann algebra $A$ generated by the model's observables. Any region $R$ of spacetime is associated with a subalgebra $A(R)\subset A$, whose observables are interpreted as being localized within that region. The algebra $A(R)$ is not necessarily type I. In other words, it's not necessarily isomorphic to an algebra of all bounded linear operators on any Hilbert space. In fact, according to section 6.5 in , there are at least some examples in which $A(R)$ is a type III factor. (A factor is a von Neumann algebra with trivial center.) According to exercise 14.4.12 in , a type III factor on a separable Hilbert space has an outer automorphism.
Even if $A(R)$ has an outer automorphism, it might not represent any symmetry of the model. However, if my shaky inferences are correct, then sometimes it does. One example is described in section V.4 in , and it is also nicely reviewed in . (I don't necessarily endorse the speculations proposed in , but it does include a nice review.) It involves the modular group (that is, the group of modular automorphisms), which is also reviewed in . Page 7 in  says "In general, the modular group... is not a group of inner automorphisms." The modular group depends on a choice of vector in the Hilbert space. (Interestingly, as explained on pages 7-8 in , the modular automorphism groups defined by different vectors are all inner-equivalent, but I think that's beside the point here.) If $R$ is a wedge region in Minkowski space, such as the wedge covered by a Rindler coordinate system, then the modular group for $A(R)$ associated with the vacuum state acts geometrically in $R$, namely as Lorentz boosts (page 248 in ). I didn't find a direct statement confirming that this particular modular group consists of outer automorphisms, but the context strongly suggests that it does, especially the context in . If that inference is correct, then this seems to be an example of a symmetry implemented by outer automorphisms of the von Neumann algebra.
I'm not sure how to reconcile this with the fact that the Lorentz group can be implemented by unitary (that is, inner) automorphisms in $A$. I suspect that the aforementioned outer automorphisms are only outer in $A(R)$, and that they can be implemented as inner automorphisms in $A$, but I'm not sure about that. Really pushing beyond the limits of my understanding here.
A couple of extra notes:
According to section V.4.2 in , "In general the wedge regions are the only ones for which the modular automorphisms (induced by the vacuum state) correspond to point transformations in Minkowski space. However, if the theory is conformally invariant there are wider classes of regions for which the modular automorphisms act geometrically. They include, as the most important case, the diamonds...''
Section 6.2 in  describes an idea relating the modular group (of presumably outer automorphisms) to the AdS/CFT correspondence, and this section together with appendix D also constitutes another nice review of the mathematical background.
 Witten (2018), "Notes on Some Entanglement Properties of Quantum Field Theory," http://arxiv.org/abs/1803.04993
 Kadison and Ringrose (1997), Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory (American Mathematical Society)
 Haag (1996), Local Quantum Physics (Springer)
 Connes and Rovelli (1994), "Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories," http://www.alainconnes.org/docs/carlotime.pdf
 Papadodimas (2013), "State-Dependent Bulk-Boundary Maps and Black Hole Complementarity," https://arxiv.org/abs/1310.6335