As pointed out by abx [here][1], the following example fits the bill. 

Take a nodal cubic in $\mathbb{P}^2$, this is an irreducible scheme proper over $\mathbb{C}$, take its normalization and throw out a point from the inverse image of the node. We get a $\mathbb{C}$-morphism from the affine line to our cubic, which is bijective on the underlying spaces. Since a non-empty proper closed subspace of either the affine line or the cubic is a finite union of closed points, this morphism is also closed, thus it induces a homeomorphism on the underlying spaces. I find it somewhat amusing that this example was not pointed out earlier (indeed, it seems to be even simpler than Julian Rosen's much upvoted example). 

  [1]: https://mathoverflow.net/q/330539/140149