We obtained [O III] narrow-band imaging and multi-slit MXU spectroscopy of the blue compact dwarf (BCD) galaxy NGC 1705 with FORS2@VLT to derive chemical abundances of PNe and H II regions and, more in general, to characterize the properties of the ionized gas. The auroral [O III]λ4363 line was detected in all but one of the eleven analyzed regions, allowing for a direct estimate of their electron temperature. The only object for which the [O III]λ4363 line was not detected is a possible low-ionization PN, the only one detected in our data. For all the other regions, we derived the abundances of Nitrogen, Oxygen, Neon, Sulfur and Argon out to ~ 1 kpc from the galaxy center. We detect for the first time in NGC 1705 a negative radial gradient in the oxygen metallicity of -0.24 ± 0.08 dex kpc^{-1}. The element abundances are all consistent with values reported in the literature for other samples of dwarf irregular and blue compact dwarf galaxies. However, the average (central) oxygen abundance, 12 + log(O/H)=7.96 ± 0.04, is ~0.26 dex lower than previous literature estimates for NGC 1705 based on the [O III]λ4363 line. From classical emission-line diagnostic diagrams, we exclude a major contribution from shock excitation. On the other hand, the radial behavior of the emission line ratios is consistent with the progressive dilution of radiation with increasing distance from the center of NGC 1705. This suggests that the strongest starburst located within the central $∼$150 pc is responsible for the ionization of the gas out to at least $∼$1 kpc. The gradual dilution of the radiation with increasing distance from the center reflects the gradual and continuous transition from the highly ionized H II regions in the proximity of the major starburst into the diffuse ionized gas. ; Comment: Accepted for publication on AJ