The answer is "no" in general, even if one asks if the classes of invertible sheaves generated $K_0(X)$ as a ring;  K3 surfaces provide a counterexample. For a K3 surface $X$ over $\mathbb{C}$, the subring of $$K_0(X)_\mathbb{Q}$$ has uncountable dimensional as a $\mathbb{Q}$-vector space; this follows, for example, from the fact that this vector space is isomorphic to the rational Chow ring of $X$ (which was proven to be very large by Mumford). But the subspace generated by line bundles has countable dimension (as the Picard group of $X$ is a finitely generated abelian group of rank at most 20).

Sorry for the very brief answer -- will try to track down references later...