What if the travelling salesman travelled by plane? It seems intuitive to solve a 2D dot-to-dot travelling salesman problem by eye using a greedy strategy. However we can only solve the TSP by eye if the graph is topographically accurate e.g. if A to B is 10 and A to C is 10 then B to C can't be 1000.

Is the greedy strategy still sub-optimal when we obey 2D scaling, ie travelling by plane? Below I managed to create a topographically accurate example where the greedy strategy is indeed sub optimal: 



Starting at S:


Greedy: S B C A S => 2.83 + 4 + 5 + 2.2 => 14.03
Optimal: S A B C S => 2.2 + 3 + 4 + 3.16 => 12.36


Is there something special about the example above that will be common to all suboptimal greedy routes? Can geometry be used to minimize error?
I suggest you read more about the Traveling Salesman Problem.  All of your questions are answered in other standard references (such as the Wikipedia link I gave).  No, the greedy algorithm is not optimal. You may be interested in the Euclidean TSP, which remains NP-hard.  The greedy algorithm is not optimal for that case, either.  No efficient algorithm is known that can always find the optimal solution, though there are algorithms that may be useful in practice.