More examples are given as answers to [a similar question][1] 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 $d \geq 4$).

  [1]: https://cstheory.stackexchange.com/questions/5251/geometric-problems-that-are-np-complete-in-r3-but-tractable-in-r2