Recent spectroscopic models of active galactic nuclei (AGNs) have indicated that the recommended electron-ion recombination rate coefficients for iron ions with partially filled M shells are incorrect in the temperature range where these ions form in photoionized plasmas. We have investigated this experimentally for Fe XIV forming Fe XIII. The recombination rate coefficient was measured by employing the electron-ion merged-beams method at the Heidelberg heavy-ion test storage ring. The measured energy range of 0-260 eV encompasses all dielectronic recombination 1s²2s²2p⁶3l3l'3l'' nl''' resonances associated with the 3p_1/2 → 3p_3/2, 3s → 3p, 3p → 3d, and 3s → 3d core excitations within the M shell of the Fe XIV (1s²2s²2p⁶3s²3p) parent ion. This range also includes the 1s²2s²2p⁶3l3l'4l'' nl''' resonances associated with 3s → 4l'' and 3p → 4l'' core excitations. We find that in the temperature range 2-14 eV, where Fe XIV is expected to form in a photoionized plasma, the Fe XIV recombination rate coefficient is orders of magnitude larger than previously calculated values.