The potential energy surfaces of the ground and low-lying excited states for the insertion reaction of atomic fluorine (F) and fluoride (F(-)) into the dilithium (Li2) molecule have been investigated. We have carried out explicitly correlated multi-reference configuration interaction (MRCI-F12) calculations using Dunning's augmented correlation-consistent basis sets. For the neutral system, the insertion of F into Li2 proceeds via a harpoon-type mechanism on the ground state surface, involving a covalent state and an ionic state which avoid each other at long distance. A detailed analysis of the changes in the dipole moment along the reaction coordinate reveals multiple avoided crossings among the excited states and shows that the charge-transfer processes play a pivotal role for the stabilization of the low-lying electronic states of Li2F. For the anionic system, which is studied for the first time, the insertion of F(-) is barrierless for many states and there is a gradual charge transfer from F(-) to Li2 along the reaction path. We also report the optimized parameters and the spectroscopic properties of the five lowest states of the neutral and seven lowest states of the anionic systems, which are strongly stabilized with respect to their respective Li2 + F/F(-) asymptotes. The observed barrierless insertion mechanisms for both systems make them good candidates for investigation under the ultracold regime.