We report the results of spectroscopic observations, obtained with the Gemini North Multi-Object Spectrograph, of nine planetary nebulae (PNe) and 15 H ii regions located in the 5.5 × 5.5 arcmin2 inner region of the nearby starburst galaxy IC10. Twelve new candidates PNe have been discovered during our pre-imaging phase. Nine of them have been spectroscopically confirmed. The direct availability of the electron temperature diagnostics in several nebulae allowed an accurate determination of the metallicity map of IC10 at two epochs: the present-time from H ii regions and the old/intermediate-age from PNe. We found a non-homogeneous distribution of metals at both epochs, but similar average abundances were found for the two populations. The derived age–metallicity relation shows a little global enrichment interpreted as the loss of metals by supernova winds and to differential gas outflows. Finally, we analysed the production of oxygen – through the third dredge-up – in the chemical abundance patterns of the PN populations belonging to several dwarf irregular galaxies. We found that the third dredge-up of oxygen is a metallicity dependent phenomenon occurring mainly for 12 + log(O/H) ≤ 7.7 and substantially absent in IC10 PNe.