This paper is concerned with the dominant pole analysis of asymptotically stable time-delay positive systems (TDPSs). It is known that a TDPS is asymptotically stable if and only if its corresponding delay-free system is asymptotically stable, and this property holds irrespective of the length of delays. However, convergence performance (decay rate) should degrade according to the increase of delays and this intuition motivates us to analyze the dominant pole of TDPSs. As a preliminary result, in this paper, we show that the dominant pole of a TDPS is always real. We also construct a bisection search algorithm for the dominant pole computation, which readily follows from recent results on α-exponential stability of asymptotically stable TDPSs. Then, we next characterize a lower bound of the dominant pole as an explicit function of delays. On the basis of the lower bound characterization, we finally show that the dominant pole of an asymptotically stable TDPS is affected by delays if and only if associated coefficient matrices satisfy eigenvalue-sensitivity condition to be defined in this paper. Moreover, we clarify that the dominant pole goes to zero (from negative side) as time-delay goes to infinity if and only if the coefficient matrices are eigenvalue-sensitive.