Heuristics for tsp without triangle inequality
I hope this partly answers your question: Most known heuristics like Greedy, Naerest Neighbor, Lin-Kerninhan etc. 
perfrom quite well for (symmetric) TSP without triangle inequality.

You may check these heuristics with the Concorde TSP Solver 
which is the best TSP solver I know so far.

In theory, there is no heuristic for TSP without triangle inequality that performs well 
(like Christofides for metric TSP). 
This is due to the well known fact that TSP cannot be approximated within any 
polynomial time computable function unless P=NP.