  We studied the electronic structure of $\beta$-Li$_2$IrO$_3$ insulator within
the density-functional theory using the generalized gradient approximation with
taking into account strong Coulomb correlations in the framework of the fully
relativistic spin-polarized Dirac linear muffin-tin orbital band-structure
method. The $\beta$-Li$_2$IrO$_3$ undergoes a pressure-induced structural and
magnetic phase transitions at $P_c$ $\sim$4 GPa with symmetry lowering to the
monoclinic $C2/c$. The structural phase transition is accompanied by the
formation of Ir$_2$ dimers on the zigzag chains, with an Ir-Ir distance of
$\sim$2.66~\AA, even shorter than that of metallic Ir. The strong dimerization
stabilizes the bonding molecular-orbital state, leads to the collapse of the
magnetism and opens the energy gap with a concomitant electronic phase
transition from a Mott insulator to band insulator. The resonant inelastic
x-ray scattering spectra (RIXS) at the Ir $L_3$ edge were investigated
theoretically from first principles. The calculated results are in good
agreement with the experimental data. We show that the the drastic
reconstruction of the RIXS spectral peak at 0.7 eV associated with the
structural $Fddd \rightarrow C2/c$ phase transition at $P_c$ can be related to
disappearing of the Coulomb correlations in the high-pressure $C2/c$ phase
