Discrete Liouville first passage percolation (LFPP) with parameter ξ> 0 is the random metric on a sub-graph of Z2 obtained by assigning each vertex z a weight of eξh(z), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all ξ> 0. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size 2 n is typically at least 2 αn for an exponent α> 0 depending on ξ. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all ξ> 0 and also has theoretical implications for the study of distances in Liouville quantum gravity.