We analyse high-resolution near-UV and optical spectra of the afterglow of GRB 080310, obtained with the Very Large Telescope Ultraviolet and Visual Echelle Spectrograph (VLT/UVES), to investigate the circumburst environment and the interstellar medium of the gamma-ray burst (GRB) host galaxy. The VLT rapid-response mode (RRM) enabled the observations to start only 13 minutes after the Swift trigger and a series of four exposures to be collected before dawn. A low neutral-hydrogen column-density (log N (HI) = 18.7) is measured at the host-galaxy redshift of z = 2.42743. At this redshift, we also detect a large number of resonance ground-state absorption lines (e.g., CII, MgII, AlII, SiII, CrII, CIV, SiIV), as well as time-varying absorption from the fine-structure levels of FeII. Time-varying absorption from a highly excited FeIII energy level (7S3), giving rise to the so-called UV34 line triplet, is also detected, for the first time in a GRB afterglow. The CrII ground-state and all observed FeII energy levels are found to depopulate with time, whilst the FeIII 7S3 level is increasingly populated. This absorption-line variability is clear evidence of ionization by the GRB, which is for the first time conclusively observed in a GRB afterglow spectrum. We derive ionic column densities at each epoch of observations by fitting absorption lines with a four-component Voigt-profile model. We perform CLOUDY photo-ionization modelling of the expected pre-burst ionic column densities, to estimate that, before the onset of the burst, [C/H] = -1.3 \pm 0.2, [O/H] < -0.8, [Si/H] = -1.2 \pm 0.2, [Cr/H] = +0.7 \pm 0.2, and [Fe/H] = +0.2 \pm 0.2 for the integrated line profile, indicating strong overabundances of iron and chromium. For one of the components, we observe even more extreme ratios of [Si/Fe] \leq -1.47 and [C/Fe] \leq -1.74. [abridged]