  We present magnetization measurements carried out on polycrystalline and
single-crystalline samples of $\alpha$-Li$_2$IrO$_3$ under hydrostatic
pressures up to 2 GPa and establish the temperature-pressure phase diagram of
this material. The N\'eel temperature ($T_{\rm{N}}$) of $\alpha$-Li$_2$IrO$_3$
is slightly enhanced upon compression with $dT_{\rm{N}}/dp$ = 1.5 K/GPa. Above
1.2 GPa, $\alpha$-Li$_2$IrO$_3$ undergoes a first-order phase transition toward
a nonmagnetic dimerized phase, with no traces of the magnetic phase observed
above 1.8 GPa at low temperatures. The critical pressure of the structural
dimerization is strongly temperature-dependent. This temperature dependence is
well reproduced on the ab initio level by taking into account lower phonon
entropy in the nonmagnetic phase. We further show that the initial increase in
$T_{\rm{N}}$ of the magnetic phase is due to a weakening of the Kitaev
interaction $K$ along with the enhancement of the Heisenberg term $J$ and
off-diagonal anisotropy $\Gamma$. Our study reveals a common thread in the
interplay of magnetism and dimerization in pressured Kitaev materials.
