Are there any problems that are NP-complete when using Euclidean geometry but are well-defined and solvable in polynomial time for some non-euclidean geometry?
Maximum TSP is polynomial time solvable under polyhedral norms, but NP-hard for Euclidean norms (optimization as well as decision version). Whether the latter is also NP-easy is a different question. (You might be able to define a somewhat artifical variant that is in NP, since the instances created for the NP-hardness proof only require bounded precision.)
A. Barvinok, S. P. Fekete, D. S. Johnson, A. Tamir, G. J. Woeginger, and R. Wodroofe. The geometric maximum Traveling Salesman problem. Journal of the ACM, 50:641-664, 2003.