  We present deep 3500$-$10000 $\AA$ spectra of H II regions and planetary
nebulae (PNe) in the starburst irregular galaxy NGC 4449, acquired with the
Multi Object Double Spectrograph at the Large Binocular Telescope. Using the
"direct" method, we derived the abundance of He, N, O, Ne, Ar, and S in six H
II regions and in four PNe in NGC 4449. This is the first case of PNe studied
in a starburst irregular outside the Local Group. Our H II region and PN sample
extends over a galacto-centric distance range of $\approx$2 kpc and spans
$\approx$0.2 dex in oxygen abundance, with average values of $12+\log(O/H)=8.37
\pm 0.05$ and $8.3 \pm 0.1$ for H II regions and PNe, respectively. PNe and H
II regions exhibit similar oxygen abundances in the galacto-centric distance
range of overlap, while PNe appear more than $\sim$1 dex enhanced in nitrogen
with respect to H II regions. The latter result is the natural consequence of N
being mostly synthesized in intermediate-mass stars and brought to the stellar
surface during dredge-up episodes. On the other hand, the similarity in O
abundance between H II regions and PNe suggests that NGC 4449' s interstellar
medium has been poorly enriched in $\alpha$ elements since the progenitors of
the PNe were formed. Finally, our data reveal the presence of a negative oxygen
gradient for both H II regions and PNe, whilst nitrogen does not exhibit any
significant radial trend. We ascribe the (unexpected) nitrogen behaviour as due
to local N enrichment by the conspicuous Wolf-Rayet population in NGC 4449.
