Maybe this "answer" is too far away from schemes to be what is desired.  But here it goes anyway.

I believe in general that for complex affine varieties $X,Y$, that a morphism $f:X\to Y$ is étale iff it is a local analytic isomorphism in the analytic topology.  When $X,Y$ are smooth, it is enough to just check that the tangent spaces are isomorphic at every point.

For example, again if I recall correctly, if a finite group $\Gamma$ acts on $X$ algebraically (preserves its coordinate ring), then the mapping $X\to X//\Gamma$ is étale. 

One should be able to find projections like that who are not homeomorphic to their image; a necessary requirement for open immersions.  The first thing that comes to mind is $\mathbb{C}^* \to \mathbb{C}^*//\mathbb{Z}_2 \cong \mathbb{C}$ where $\mathbb{Z}_2$ acts by $z\mapsto 1/z$.

Conversely however, open immersions are always étale.