We develop a theory of finite-temperature momentum-resolved tunneling spectroscopy (MRTS) for disordered, interacting, two-dimensional, topological-insulator edges. The MRTS complements conventional electrical transport measurement in characterizing the properties of the helical Luttinger liquid edges. Using the standard bosonization technique, we study low-energy spectral function and the MRTS tunneling current, providing a detailed description controlled by disorder, interaction, and temperature, taking into account Rashba spin-orbit coupling, interedge interaction, and distinct edge velocities. Our theory provides a systematic description of the spectroscopic signals in the MRTS measurement we hope will stimulate future experimental studies on the two-dimensional time-reversal invariant topological insulator.