Context. The Tarantula Nebula (a.k.a. 30 Dor) is a spectacular star-forming region in the Large Magellanic Cloud (LMC), seen through gas in the Galactic disc and halo. Diffuse interstellar bands (DIBs) offer a unique probe of the diffuse, cool-warm gas in these regions. Aims: The aim is to use DIBs as diagnostics of the local interstellar conditions, whilst at the same time deriving properties of the yet-unknown carriers of these enigmatic spectral features. Methods: Spectra of over 800 early-type stars from the Very Large Telescope Flames Tarantula Survey (VFTS) were analysed. Maps were created, separately, for the Galactic and LMC absorption in the DIBs at 4428 and 6614 Å and - in a smaller region near the central cluster R 136 - neutral sodium (the Na i D doublet); we also measured the DIBs at 5780 and 5797 Å. Results: The maps show strong 4428 and 6614 Å DIBs in the quiescent cloud complex to the south of 30 Dor but weak absorption in the harsher environments to the north (bubbles) and near the OB associations. The Na maps show at least five kinematic components in the LMC and a shell-like structure surrounding R 136, and small-scale structure in the Milky Way. The strengths of the 4428, 5780, 5797 and 6614 Å DIBs are correlated, also with Na absorption and visual extinction. The strong 4428 Å DIB is present already at low Na column density but the 6614, 5780 and 5797 Å DIBs start to be detectable at subsequently larger Na column densities. Conclusions: The carriers of the 4428, 6614, 5780 and 5797 Å DIBs are increasingly prone to removal from irradiated gas. The relative strength of the 5780 and 5797 Å DIBs clearly confirm the Tarantula Nebula as well as Galactic high-latitude gas to represent a harsh radiation environment. The resilience of the 4428 Å DIB suggests its carrier is large, compact and neutral. Structure is detected in the distribution of cool-warm gas on scales between one and >100 pc in the LMC and as little as 0.01 pc in the Sun's vicinity. Stellar winds from the central cluster R 136 have created an expanding shell; some infalling gas is also detected, reminiscent of a galactic "fountain". Full Tables A.2-A.4 are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/550/A108