Context. Luminous infrared galaxies (LIRGs) are an important class of objects in the low-z universe bridging the gap between normal spirals and the strongly interacting and starbursting ultraluminous infrared galaxies (ULIRGs). Since a large fraction of the stars in the Universe have been formed in these objects, LIRGs are also relevant in a high-z context. Studies of the two-dimensional physical properties of LIRGs are still lacking. Aims: We aim to understand the nature and origin of the ionization mechanisms operating in the extra-nuclear regions of LIRGs as a function of the interaction phase and infrared luminosity. Methods: This study uses optical integral field spectroscopy (IFS) data obtained with VIMOS. Our analysis is based on over 25 300 spectra of 32 LIRGs covering all types of morphologies (isolated galaxies, interacting pairs, and advanced mergers), and the entire 1011-1012 L&sun; infrared luminosity range. Results: We found strong evidence for shock ionization, with a clear trend with the dynamical status of the system. Specifically, we quantified the variation with interaction phase of several line ratios indicative of the excitation degree. While the [N ii]λ6584/Hα ratio does not show any significant change, the [S ii]λλ6717,6731/Hα and [O i]λ6300/Hα ratios are higher for more advanced interaction stages. Velocity dispersions are higher than in normal spirals and increase with the interaction class (medians of 37, 46, and 51 km s-1 for class 0-2, respectively). We constrained the main mechanisms causing the ionization in the extra-nuclear regions (typically for distances ranging from ~0.2-2.1 kpc to ~0.9-13.2 kpc) using diagnostic diagrams. Isolated systems are mainly consistent with ionization caused by young stars. Large fractions of the extra-nuclear regions in interacting pairs and more advanced mergers are consistent with ionization caused by shocks of vs ⪉ 200 km s-1. This is supported by the relation between the excitation degree and the velocity dispersion of the ionized gas, which we interpret as evidence for shock ionization in interacting galaxies and advanced mergers but not in isolated galaxies. This relation does not show any dependence with the infrared luminosity (i.e. the level of star formation). All this indicates that tidal forces play a key role in the origin of the ionizing shocks in the extra-nuclear regions. We also showed for the first time what appears to be a common log([O i]λ6300/Hα) - log (σ) relation for the extranuclear ionized gas in interacting (U)LIRGs (i.e. covering the entire 1011.0-1012.3 L&sun; luminosity range). This preliminary result needs to be investigated further with a larger sample of ULIRGs. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (ESO Programs 076.B-0479(A), 078.B-0072(A) and 081.B-0108(A)).Figures 14-17 are only available in electronic form at http://www.aanda.org