In this paper, we propose a general approach based on local search and incremental preference elicitation for solving multi-objective combinatorial optimization problems with imprecise preferences. We assume that the decision maker's preferences over solutions can be represented by a parameterized scalarizing function but the parameters are initially not known. In our approach, the parameter imprecision is progressively reduced by iteratively asking preference queries to the decision maker 1) before the local search in order to identify a promising starting solution and 2) during the local search but only when preference information are needed to discriminate between the solutions within a neighborhood. This new approach is general in the sense that it can be applied to any multi-objective combinatorial optimization problem provided that the scalarizing function is linear in its parameters (e.g., a weighted sum, an OWA aggregator, a Choquet integral) and that a (near-)optimal solution can be efficiently determined when preferences are precisely known. For the multi-objective traveling salesman problem, we provide numerical results obtained with different query generation strategies to show the practical efficiency of our approach in terms of number of queries, computation time and gap to optimality.