Assume that ${\rm char}=0$. If $X$ is Fano and $\mathcal L$ is an ample invertible sheaf contained in, but not equal to $\omega_X^{-1}$, then $H^i(X,\mathcal L^{-1})=0$ for all $i$. This is Kodaira vanishing for $i\neq \dim X$ and for $n=\dim X$, $H^n(X,\mathcal L^{-1})$ is dual to $H^0(X, \mathcal L\otimes \omega_X)$ which is $0$ by the choice of $\mathcal L$.

I think both of your examples are covered by this.

In general, if $\mathcal L$ is contained in $\omega_X$, then $H^n(X,\mathcal L)\neq 0$, so for a K3 (or more generally if $K_X=0$) negative ample invertible sheaves will not work. It is also relatively easy to see that if ${\rm Pic}\, X = \mathbb Z$, then there are no such invertible sheaves.

**Edit**:
On the other hand, it is relatively easy to find invertible sheaves with this property on K3's with higher Picard numbers. For instance, let $X$ be a K3 and $L_1, L_2$ two disjoint $(-2)$-curves. This requires the Picard number to be at least $3$, but then it is quite common. For instance, every Kummer has plenty of disjoint $(-2)$-curves. So, let $\mathcal L=\mathcal O_X(L_1-L_2)$. Then $\mathcal L$ has no cohomology for any $i$.

This can be seen the following way: First notice that the sheaf $\mathcal O_X(L_1)$ has a $1$-dimensional $H^0$ since $L_1$ has negative self-intersection, $H^2=0$ since its dual has no $H^0$, and $H^1=0$ by Riemann-Roch. Then in the short exact sequence
$$
0 \to \mathcal L \to \mathcal O_X(L_1) \to \mathcal O_{L_2} \to 0
$$
the two sheaves on the right have the exact same cohomology groups and clearly the induced map on $H^0$ is an isomorphism, so the sheaf on the left cannot have any non-zero cohomology.

**Second edit:** I just realized that this can be done much simpler: $H^0(X,\mathcal L)=0$ since if it had a non-zero section, that would produce an effective divisor linearly equivalent to $L_1$, but not equal to it. Similarly $h^2(X,\mathcal L^{-1})=h^0(X, \mathcal O_X(L_2-L_1))=0$. Finally, then Riemann-Roch shows that $h^1=0$ as well. This argument has the advantage that it shows that if $L_1,L_2$ intersect, then $h^1\neq 0$.

Using similar ideas one can make up lots of invertible sheaves on K3 surfaces with this property.