NP-complete graph property that is hereditary, but not additive?
I think the $k$-clique cover problem, which asks whether there exists a partition of the vertices in $k$ sets such that each set induces a clique, has the desired properties.

Clearly, taking induced subgraphs can't make the minimum size of such partition increase. On the other hand, when you take the disjoint union of two graphs, you have to take the union of the partition into cliques of each one.