(This started as a comment but then got a bit too long.)
In brief: I'd be interested to know the answer myself. But it would be a (pleasant) surprise to me if there is something exactly of that sort covering a sizable portion of Hodge theory.
Other sources, apart from GH and Voisin, which may be useful are Demailly's Complex analytic and algebraic geometry (exercises on Hodge theory promised in the table of contents but actually absent in the version I have) and Kulikov and Kurchanov's Complex algebraic varieties (Springer EMS, Algebraic geometry 3).
I think one can extract a couple of problems from every chapter of these; if you are looking for computations, you might want to take a look at e.g. VI.10 of Demailly (complex curves, Abel-Jacobi map, Weierstrass points).
Kulikov and Kurchanov have examples of algebraic non-projective and Moishezon non-algebraic varieties (1.3; strictly speaking, not a part of Hodge theory, but good to know), Kaehler non-Hodge tori (end of 1.7); one could extract a couple of problems from chapter 3 on Torelli theorems. Depending on who the intended audience is, one can present a part of the proof of the non-rationality of a cubic threefold as a series of problems.
Finally, Shafarevich's Basic algebraic geometry has about a dozen exercises on Hodge theory in VIII.4. Again, if you allow for not directly related but good-to-know things (e.g. examples of surfaces isomorphic in the analytic but not algebraic category), then the whole of chapter VIII there can be useful.