Geometric problems that are NP-complete in $R^3$ but tractable in $R^2$?
It's not quite what you ask, because the 3d version is even harder than NP-complete, but: Finding a shortest path between two points among polygonal obstacles in the plane is in polynomial time (most simply, construct the visibility graph of the two terminals and the polygon vertices and apply Dijkstra; there is also a more complicated O(n log n) algorithm due to Hershberger and Suri, SIAM J. Comput. 1999) but finding a shortest path among polyhedral obstacles in 3d is PSPACE-complete (Canny and Reif, FOCS 1987).