[Fek99] showed that the maximum TSP can be solved in time O(n) for rectilinear distances in the plane, but is NP-hard for Euclidean distances in three-dimensional space, or on the surface of a sphere. Conjectures the case of planar Euclidean distances to be NP-hard.
More recent details and related problems can be found in [BFJ+03].
Also related is the Planar Euclidean maximum scatter TSP: What is the complexity of finding a tour for a planar point set in ℜ^{d}, such that the Euclidean length of the shortest edge is maximized? Stated in [ACM+97], shown NP-hard in dimensions d≥3 in [Fek99], open for d = 2. Also, no bounds on approximation are known in a geometric context; the best known aproximation algorithm from [ACM+97] achieves an approximation factor of 2, but does not use geometry.