An excitonic insulator (EI) is a correlated-electron state in which excitons condense into an insulating ground state in analogy with the condensation of Cooper pairs in a superconductor. A handful of EI candidates have been proposed, but to date experimental verification of such a state has not been conclusively found. Recent transport and specific heat measurements have placed Ta₂NiSe₅ as the leading EI candidate with T_c = 326 K. Here we present measurements of the terahertz and far-infrared optical conductivity of Ta₂NiSe₅, which show that at temperatures far below T_c a thermally-activated Drude peak gradually develops inside the optical band gap E_g^op = 160 meV. At the same time, the gap gradually fills from the high-frequency side with significant spectral weight from above-gap states. This transfer of spectral weight can be interpreted as predominant incoherent hopping of charge carriers along the Ta-Ni chains in the orthorhombic phase and as evidence for near-zero-gap behavior at high temperature. A phonon centered at 4.71 meV in both the a- and c-axis response suggests that the monoclinic distortion associated with the formation of the EI state allows bidirectional ac-plane activity of the B_1u/B_3u modes.