There is
More examples are given as answers to a similar question about problems NP-hard in $\mathbb R^3$ but not in $\mathbb R^2$:
- Set-cover by half-spaces.
- Finding a shortest path between two points among polygonal obstacles.
- Determining whether a non-convex polygon/polyhedron can be triangulated without Steiner points.
- Realizability problem for $d$-dimensional polytopes is a candidate ($d \leq 3$ vs $d \geq 4$).
