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Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?

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No. You are free to define viscosity solutions for any order equation (by touching with test functions), but uniqueness of viscosity solutions is based on the maximum principle (or rather, the comparison principle), which limits the usefulness to second order equations. Without comparison you lose all the useful properties of viscosity solutions (uniqueness, stability under weak limits, the Perron method, etc.).

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