The nuclear magnetic resonance (NMR) shift, χα, at low temperatures is examined for massless Dirac electrons in the organic conductor, α-(BEDT-TTF)2I3, where α [= A (= AA), B, and C] denotes the sites of the four molecules in the unit cell. The Dirac cone exists within an energy of 0.01 eV between the conduction and valence bands. The magnetic response function is calculated by taking account of the long-range Coulomb interaction and electron doping. Calculating the interaction within the first order in the perturbation, the chemical potential is determined self-consistently, and the self-energy and vertex corrections are taken to satisfy the Ward identity. The site-dependent χα is calculated at low temperatures of 0.0002 &lt
T &lt
0.002 (T is temperature in the unit of eV) by correctly treating the wave function of the Dirac cone. At lower (higher) temperatures the self-energy (vertex) correction of χα at all sites except for B is dominant and the sign is negative (positive), while the sign of the correction at the B site is always negative. For moderate doping, the shift as a function of T takes a minimum at which χC ≃ χA = χA′ &gt
χB. The relevance of the shift to the experiment is discussed.