We introduce an extended Birnbaum component importance measure considering epistemic and aleatory uncertainty adapted to non-coherent systems. The belief function theory is proposed as a framework for taking into account both types of uncertainty. The objective is to rank components according to their importance in system working. This importance measure was introduced for coherent systems; however, the increasing complexity of modern systems introduces the case of non-coherent systems. This is why we should consider these kinds of systems. In this work, we propose a method to compute the importance measure of the components of non-coherent systems in the framework of belief functions theory.