Here we study 16 planetary nebulae (PNe) in the dwarf irregular galaxy NGC 205 by using Gemini Multi-Object Spectrographs (GMOS@Gemini) spectra to derive their physical and chemical parameters. The chemical patterns and evolutionary tracks for 14 of our PNe suggest that there are no type I PNe among them. These PNe have an average oxygen abundance of 12+log(O/H) = 8.08 ± 0.28 and progenitor masses of 2-2.5 M⊙ and thus were born ˜1.0-1.7 Gyr ago. Our results are in good agreement with previous PN studies of NGC 205. The present 12+log(O/H) is combined with our previous work and results in the literature to study the PN metallicity trends of Local Group (LG) dwarf galaxies, in an effort to establish the PN luminosity-metallicity and mass-metallicity relations (LZR and MZR) for LG dwarf irregulars (dIrrs) and dwarf spheroidals (dSphs). Previous attempts to obtain such relations failed to provide correct conclusions because they were based on limited samples. As far as we are able to compare stellar with nebular metallicities, our MZR is in very good agreement with the slope of the MZR recently obtained for LG dwarf galaxies using spectroscopic stellar metallicities by Kirby and coauthors. In fact, we found that both dIrr and dSph galaxies follow the same MZR, at variance with the differences claimed in the past. Moreover, our MZR is also consistent with the global MZR of star-forming galaxies, which span a wider stellar mass range (˜106-1011 M⊙).