List of NP-hard problems, where there is active research in practical heuristics I am looking for list of NP-hard optimization problems, where there is active research in practical heuristic for solving them and there are common benchmarks, which people try to beat.

Examples include:
Phylogenetic tree reconstruction (heuristic for example here)
Travelling salesman (not so active, but LKH is quite well known)

More specifically, I am looking for areas of research, where people really care about resulting cost (like TSP or phylogeny mentioned above). 
E.g. finding decision tree is not a thing I am looking for, since very few people care about resulting tree height.
MaxSAT - people actually care about this because SAT solvers are so well-developed that often the best route for your favorite NP optimization problem in practice is to reduce it to MaxSAT and then apply one of the well-known solvers. Check out the SAT competition for benchmarks etc.

Clique-finders get used in computational biology and combinatorics, and the heuristic algorithms are shockingly good, as I recall.

Vast portions of Operations Research is devoted to algorithms, including heuristic ones, for solving cases of integer or mixed-integer linear programming.