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We use VLT/UVES high-resolution optical spectroscopy of seven GRB afterglows (at z_GRB between 2.20 and 3.97) to investigate circumburst and interstellar plasma in the host galaxies of the bursts. GRBs 050730, 050820, 050922C, 060607, 071031, and 080310 were each detected by the Swift satellite. With the minimum possible time delay (as short as eight minutes), follow-up optical spectroscopy at 6.0km/s resolution began with UVES. We present Voigt-profile component fits and analysis of the high-ion absorption detected in these spectra and in the pre-Swift UVES spectrum of GRB 021004, at velocities between z_GRB-5000km/s and z_GRB. We identify several distinct categories of high-ion absorption at velocities close to z_GRB: (i) strong high-ion absorption at z_GRB itself is always seen in OVI, CIV, and SiIV, usually (in six of seven cases) in NV, and occasionally in SIV and SVI. Three of the cases show logN(NV)>14 in a single-component, suggesting a circumburst origin, but we cannot rule out an interstellar origin. Indeed, using the non-detection of SIV^* at z_GRB toward GRB 050730 together with a UV photo-excitation model, we place a lower limit of 400pc on the distance of the SIV-bearing gas from the GRB; (ii) complex, multi-component CIV and SiIV profiles extending over 100-400km/s around z_GRB are observed in each spectrum; these velocity fields are similar to those measured in CIV in damped Lyman-α systems at similar redshifts, suggesting a galactic origin; (iii) asymmetric, blueshifted, absorption-line wings covering 65-140km/s are seen in the CIV, SiIV, and OVI profiles in four of the seven GRB afterglow spectra. The wing kinematics together with the ``Galactic'' CIV/SiIV ratios measured in two cases suggest that the wings trace outflowing interstellar gas in the GRB host galaxies; (iv) high-velocity (HV; 500-5000km/s relative to z_GRB) components are detected in six of the seven spectra; these components are not necessarily a single homogeneous population, and many of them may arise in unrelated foreground galaxies. However, in the cases of GRBs 071031 and 080310, the ionization properties of the HV components (very high CIV/SiIV ratios and absence of neutral-phase absorption in SiII or CII) are suggestive of a circumburst origin; models of Wolf-Rayet winds from the GRB progenitors can explain both the kinematics and ionization level of these HV components.