  We present magnetization measurements on polycrystalline
$\beta$-Li$_2$IrO$_3$ under hydrostatic pressures up to 3~GPa and construct the
temperature-pressure phase diagram of this material. Our data confirm that
magnetic order breaks down in a first-order phase transition at $p_{\rm{c}}$
$\approx$ 1.4~GPa and additionally reveal a step-like feature -- magnetic
signature of structural dimerization -- that appears at $p_{\rm{c}}$ and shifts
to higher temperatures upon further compression. Following the structural study
by L. S. I. Veiga et al. [Phys. Rev. B 100, 064104 (2019)], we suggest that a
partially dimerized phase with a mixture of magnetic and non-magnetic Ir$^{4+}$
sites develops above $p_{\rm{c}}$. This phase is thermodynamically stable
between 1.7 and 2.7~GPa according to our ab initio calculations. It confines
the magnetic Ir$^{4+}$ sites to weakly coupled tetramers with a singlet ground
state and no long-range magnetic order. Our results rule out the formation of a
pressure-induced spin-liquid phase in $\beta$-Li$_2$IrO$_3$ and reveal
peculiarities of the magnetism collapse transition in a Kitaev material. We
also show that a compressive strain imposed by the pressure treatment of
$\beta$-Li$_2$IrO$_3$ enhances signatures of the 100~K magnetic anomaly at
ambient pressure.
