We studied the doping and temperature (T) dependence of the infrared (IR) response of Ba(Fe1-xCox)2As2 single crystals. We show that a weak band around 1000 cm-1, that was previously interpreted in terms of interaction of the charge carriers with magnetic excitations or of a pseudogap, is rather related to low-energy interband transitions. Specifically, we show that this band exhibits a similar doping and T-dependence as the hole pockets seen by angle resolved photoemission spectroscopy (ARPES). Notably, we find that it vanishes as a function of doping near the critical point where superconductivity is suppressed in the overdoped regime. Our IR data thus provide bulk specific information (complementary to the surface sensitive ARPES) for a Lifshitz transition. Our IR data also reveal a second low-energy band around 2300 cm-1 which further emphasizes the necessity to consider the multiband nature of these iron arsenides in the analysis of the optical response.